There are various ideas in the commentaries that actually discuss the grape (non-alcoholic) point (bold added to emphasize the main points).
From Thomas Whitelaw, “Introductory Essay on the Authenticity and Authorship of the Book of Numbers,” in Numbers, edited by H. D. M. Spence, Joseph S. Exell, The Pulpit Commentary (London: Funk & Wagnalls Company, 1910), pp. 50-51—
- Abstinence from the fruit of the vine. It was to be a rigorous abstinence. This we may take to signify a protest in the most comprehensive way against all seeking of mere pleasure and comfort. The grape was the symbol of sensual delights. The spies brought back grapes of Eshcol more than any other produce to testify the riches of Canaan: this shows how much the Israelites thought of the fruit. There was, of course, no peculiar merit and advantage in abstaining from the grape itself. The abstinence was simply a sign indicating a desire to rise above the common pleasures of men. The Nazarites were not ascetics. They did not refrain from a good creature of God by way of penance. But in the grape there was the possibility of wine and strong drink, and the wine and strong drink were the testimony of the worldly soul that he loved to gratify his sensual nature, and cared not that his body should be so disciplined and restrained as to be the effectual minister of God. The appropriate joys of human life are not to be found among the powers that link us to the lower creation. We are to look for them in communion with God and following Christ. Our joy is in the Holy Ghost. “Is any merry, let him sing Psalms.”
So for Whitelaw, it was simply a testimony to being committed to God over and above the fruit of the vine that brought such varied pleasures for humanity's consumption.
From Carl Friedrich Keil and Franz Delitzsch, Commentary on the Old Testament (Peabody, MA: Hendrickson, 1996), Vol. 1, pp.672-673&mdash
The vow consisted of the three following points, vv. 1–4: In the first place, he was to abstain from wine and intoxicating drink (shecar, see Lev. 10:9); and neither to drink vinegar of wine, strong drink, nor any juice of the grape (lit., dissolving of grapes, i.e., fresh must pressed out), nor to eat fresh grapes, or dried (raisins). In fact, during the whole period of his vow, he was not to eat of anything prepared from the vine, “from the kernels even to the husk,” i.e., not the smallest quantity of the fruit of the vine. The design of this prohibition can hardly have been, merely that, by abstaining from intoxicating drink, the Nazarite might preserve perfect clearness and temperance of mind, like the priests when engaged in their duties, and so conduct himself as one sanctified to the Lord (Bähr); but it goes much further, and embraces entire abstinence from all the deliciae carnis by which holiness could be impaired. Vinegar, fresh and dried grapes, and food prepared from grapes and raisins, e.g., raisin-cakes, are not intoxicating; but grape-cakes, as being the dainties sought after by epicures and debauchees, are cited in Hos. 3:1 as a symbol of the sensual attractions of idolatry, a luxurious kind of food, that was not in harmony with the solemnity of the worship of Jehovah. The Nazarite was to avoid everything that proceeded from the vine, because its fruit was regarded as the sum and substance of all sensual enjoyments.
So Keil & Delitzsch consider it a symbol of abstaining from all "deliciae carnis" (sensual pleasures), which he believes the grapes then were a "luxurious kind of food, not in harmony with the solemnity of the worship of Jehovah" (but in contrast to that, see Cole's commentary below).
From John Peter Lange, Philip Schaff, Samuel T. Lowrie, and A. Gosman, A Commentary on the Holy Scriptures: Numbers (1879; Bellingham, WA: Logos Bible Software, 2008), pp. 40-41—
- He shall separate himself from wine, etc., vers. 3, 4. The primary object of this prohibition is already intimated in the history of Aaron’s sons who were destroyed. Theocratic enthusiasm must as strictly as possible be preserved pure from all disturbance by the spirit of drunkenness. Hence the prohibition not only of wine and of all spirituous, strong drink, not only of flat wine, wine or other vinegar, but even of grape juice just expressed (מִשְׁרַת). The prohibition is symbolically intensified and completed by forbidding the enjoyment of fresh and even of dried grapes (raisins). KEIL’S notion only obscures the simple, fundamental thought, when he says that the prohibition to use grapes looks to abstinence from all deliciæ carnis so damaging to sanctification. The grape confections of Hosea 3:1 hardly serve to prove this.
For Lange, the grape and raisins were merely "symbolically" adding to the alcoholic prohibition (what could be produced by them); so he attempts to keep the focus on the alcoholic aspects; he rejects explicitly Keil's broadening to all sensual pleasures.
From Victor Harold Matthews, Mark W. Chavalas, and John H. Walton, The IVP Bible Background Commentary: Old Testament, electronic ed. (Downers Grove, IL: InterVarsity Press, 2000), Numbers 6:3-4—
Abstinence from grape products. Prohibition of grape products has suggested to some interpreters that a nomadic lifestyle is being elevated, but it is very difficult to see that as a biblical or priestly agenda. Alternatively one must notice that the grape is one of the principal, one could say characteristic, staples of Canaan and therefore symbolically connected to the issue of fertility (note that the spies bring back a huge cluster of grapes [13:24] as evidence of the fertility of the land). The use of raisins in raisin cakes for the fertility cult can be seen in Hosea 3:1.
So without explicitly saying it, Matthews et al. are arguing that such association to "fertility cult" practices was in view.
From R. Dennis Cole, Numbers, Vol. 3B, The New American Commentary (Nashville: Broadman & Holman Publishers, 2000), p. 122—
All forms of intoxicating beverage are off limits at all times to the Nazirite for the duration of the vow. This restriction is more extensive than the prohibition placed upon priests, who are limited from consumption of such drinks only during the period of tabernacle or temple service (Lev 10:9). Yet not only is a Nazirite restricted from consuming wine (yayin) and fermented drink (šēkār), but that individual also cannot partake of wine vinegar (ḥōmeṣ yayin), vinegar from other fermented liquids (ḥōmeṣ šēkār), unfermented grape juice, grapes, raisins, grape seeds, and hulls, or anything else derived from the vineyard. The reference to seeds and hulls is probably hyperbolic, emphasizing the total abstinence from the vineyard. The vineyard restriction is paralleled by the Rechabite tradition that forbade the planting of vineyards, an indication of a sedentary lifestyle. Abstaining from the vineyard and related products was a personal and generally private act of special devotion of one’s life and mind to the Lord.
The vineyard and its produce thus can have an antithetical usage in the Bible. On one hand vineyards are evidence of Yahweh’s great blessing upon the land (Isa 5:1–2, 7a; Jer 2:21). A large cluster of grapes was brought back by the team of spies who explored the land of Canaan prior to Israel’s rejection of the land (Num 13:23–24). Wine is combined with various elements in the sacrificial system for worshiping God and making atonement (Num 15:5, 7, 10; 28:7–10, 14). Israelites living far from Jerusalem were even encouraged to purchase wine and strong drink along with sheep and cattle with money from their tithes, and then they would eat and drink these in the presence of the Lord with rejoicing (Deut 14:24–27). However, excessive consumption is condemned categorically (Prov 20:1; 23:30–31; 31:4; Isa 28:7).
So notice in the second paragraph how Cole gives evidence against Keil's hypothesis noted above that wine was "luxurious kind of food, not in harmony with the solemnity of the worship of Jehovah." Cole's main emphasis is similar to Whitelaw's above, that such abstinence related primarily to showing the devoutness to God.
The fact is that the primary "reason" for the abstinence from all things related to the vine is that God said so. He made the declaration (v.1) that these things are what He expected for a vow of separation, which included this extensive abstinence from products of the vine (v.3-4). Similar to the laws of what is clean or unclean, there is not necessarily anything inherent in the objects that make them so, but simply the declaration of God that such is the case at any particular time (e.g. Acts 10:9-16).
Once the vow was over, "When the days of his separation are fulfilled" (v.13) and the Nazirite performs the finishing sacrifices of the vow (vv.14-20a), then wine could be partaken of again (and hence, all products of the vine), as v.20b makes explicit, "... After that the Nazirite may drink wine."
So the prohibition from all fruit of the vine clearly has everything to do with the Nazirite being separated to God, devoted to this vow, and not specifically to anything inherent in the grape itself. While alcohol could impair judgement (during the vow), and so had a more inherent, practical aspect to finishing the vow without being out of one's right mind of judgment, the three main points served to distinguish the one under a Nazirite vow from all others: men and women could normally partake of the fruit of the vine, so the Nazirite would stand out in social functions where people were consuming such; men and women would normally at least "trim" their hair to keep it neat, if not in some cases fully cut it off, but the Nazirite would stand out in not cutting it at all (and the hair was intended as part of the sacrifice at the end, v.18); men and women could normally take care of their dead relatives, allowing that uncleaness to come and then cleansing themselves from that (Num 19:11-13), but even that was something the Nazirite was unable to do.
So all the points, in context, relate to showing a distinction; that one has separated to God for the purpose of the vow. There are other causes of uncleanness God could have emphasized; indeed, He could have stated any conditions He wanted. He chose to state these conditions, without explicitly saying why, other than they were the distinction of being separated. There may be some relation to the fact that it is the "head" that is emphasized in the separation, for even if one accidentally ended up defiled by the dead, it "defiles his consecrated head" (v.9) and he/she was to shave off what hair was grown out. So if there is any inherent aspect to the grape prohibition, it likely relates to the consecration of one's head as well. That alcohol affects the "head" is clear; but that anything consumed occurs through the mouth (and thus "head") may indicate that God simply wanted the "head" separated from all products of the vine during this time. The "why" for the non-alcoholic forms is not given, and all speculations must remain that, speculation. (One speculation would be that the only prophecy related to grapes and vines as of the giving of this law of the Nazirite in Numbers is in relation to Judah's prophesied kingship in Genesis 49:11; so one could speculatively argue that the prohibition here shows a symbolic submission to the work of the "Him" to whom "the obedience of the people" would come in Gen 49:10, to which the vine and grapes are tied in v.11; but that is still speculation.)
And so it seems of the commentaries noted above, given what is stated in the passage in Numbers 6 itself, that Whitelaw and Cole fall the most in line with the idea that the prohibition of all fruit of the vine simply relates to showing devoutness to God by the vow taken to God. The prohibition is a sign of the devotion, nothing more.