The Early Christian Doctrine of the Eucharist
First we have to recognize and concede that Malachi 1 seen as a prophecy of the Eucharist as a sacrifice offered by the Gentiles and the Eucharist being seen as a sacrifice are not dependant at all (rather, Malachi 1 being seen as a prophecy of such is proof enough that the belief predates citations used to prove it is prophesied). So Malachi 1 or no Malachi 1, the Christians still believed the Eucharist was a sacrifice offered by the Gentiles.
Consider the teaching of St. Paul on the fundamental nature of the Eucharist, and a significant Scriptural term he uses in reference thereto:
1 Corinthians 10:14-21 (DRB)
14 Wherefore, my dearly beloved, fly from the service of idols. 15 I speak as to wise men: judge ye yourselves what I say. 16 The chalice of benediction, which we bless, is it not the communion of the blood of Christ? And the bread, which we break, is it not the partaking of the body of the Lord? 17 For we, being many, are one bread, one body, all that partake of one bread. 18 Behold Israel according to the flesh: are not they, that eat of the sacrifices, partakers of the altar? 19 What then? Do I say, that what is offered in sacrifice to idols, is any thing? Or, that the idol is any thing? 20 But the things which the heathens sacrifice, they sacrifice to devils, and not to God. And I would not that you should be made partakers with devils. 21 You cannot drink the chalice of the Lord, and the chalice of devils: you cannot be partakers of the table of the Lord, and of the table of devils.
Even without the Scriptural context of the words, we know that the table of demons refers to partaking in sacrifices offered to idols, and so the Eucharist is implicitly called a sacrifice by its being used in parallel thereto. But we're not left without Scriptural context.
"The table of the Lord" refers to the altar of sacrifice (hence its use for both the Eucharist and the things offered to idols in parallel). It only occurs in Malachi, interestingly, in the whole Bible (and in St. Paul's teaching on the Eucharist above):
Malachi 1:6-14 (DRB)
6 The son honoureth the father, and the servant his master: if then I be a father, where is my honour? and if I be a master, where is my fear? saith the Lord of hosts. 7 To you, O priests, that despise my name, and have said: Wherein have we despised thy name? You offer polluted bread upon my altar, and you say: Wherein have we polluted thee? In that you say: The table of the Lord is contemptible. 8 If you offer the blind for sacrifice, is it not evil? and if you offer the lame and the sick, is it not evil? offer it to thy prince, if he will be pleased with it, or if he will regard thy face, saith the Lord of hosts. 9 And now beseech ye the face of God, that he may have mercy on you, (for by your hand hath this been done,) if by any means he will receive your faces, saith the Lord of hosts. 10 Who is there among you, that will shut the doors, and will kindle the fire on my altar gratis? I have no pleasure in you, saith the Lord of hosts: and I will not receive a gift of your hand. 11 For from the rising of the sun even to the going down, my name is great among the Gentiles, and in every place there is sacrifice, and there is offered to my name a clean oblation: for my name is great among the Gentiles, saith the Lord of hosts. 12 And you have profaned it in that you say: The table of the Lord is defiled: and that which is laid thereupon is contemptible with the fire that devoureth it. 13 And you have said: Behold of our labour, and you puffed it away, saith the Lord of hosts, and you brought in of rapine the lame, and the sick, and brought in an offering: shall I accept it at your hands, saith the Lord? 14 Cursed is the deceitful man that hath in his flock a male, and making a vow offereth in sacrifice that which is feeble to the Lord: for I am a great King, saith the Lord of hosts, and my name is dreadful among the Gentiles.
From St. Paul's Jewish perspective enunciated in 1 Corinthians 10, it is also to be gathered that Malachi 1 cannot be taken to mean God approves of sacrifices "offered to devils"—which is what "the heathens offer." Much less that they are "a pure oblation [sacrificial offering]!"
The Present Tense in Prophecy
Early Christians virtually without exception used the Septuagint Old Testament, not the Hebrew. As such, they read:
Malachi 1:11 (LXX) διότι ἀπ' ἀνατολῶν ἡλίου ἕως δυσμῶν τὸ ὄνομά μου δεδόξασται ἐν τοῖς ἔθνεσιν καὶ ἐν παντὶ τόπῳ θυμίαμα προσάγεται τῷ ὀνόματί μου καὶ θυσία καθαρά διότι μέγα τὸ ὄνομά μου ἐν τοῖς ἔθνεσιν λέγει κύριος παντοκράτωρ
Malachi 1:11 (Brenton) For from the rising of the sun even to the going down thereof my name has been glorified among the Gentiles; and in every place incense is offered to my name, and a pure offering: for my name is great among the Gentiles, saith the Lord Almighty.
So Christians knew their Greek but they still viewed it as prophetic. Much as with "unto us a child is born, unto us a son is given" (Isaiah 9:6). The verbless phrase "διότι μέγα τὸ ὄνομά μου ἐν τοῖς ἔθνεσιν" ("for great [is] my name among the Gentiles"), a grecanized form of the Hebrew, as far as I know, cannot be read in a literal sense to refer to the future, either. This is a case where a prophecy is said as though it were a sure, past fact, but has reference to the future.
A Jewish Rabbi of the 12th century once wrote concerning this kind of prophecy:
And you should know that it is a typical behavior of the past tense verbs in the holy language to use a past tense verb in place of a future tense verb (which are the letters איתן), and this is mostly in prophecies because the matter is clear as if it passed, because it has already been decreed.1
And (on Psalm 3:5):
Or the holy spirit expressed itself in his language, and this is the correct explanation, for we have already explained that all the Psalms were [transmitted] through the holy spirit, and in most cases of prophecy the narrator speaks in past tense, in the place of future tense, as though the deed is already done.2
Such seems to have been the understanding of such early Christians as Justin Martyr, who does not cite it in the future tense (paraphrasing it) but views it as foreknowledge (in concordance with Radak's statement above):
"And the offering of fine flour, sirs," I said, "which was prescribed to be presented on behalf of those purified from leprosy, was a type of the bread of the Eucharist, the celebration of which our Lord Jesus Christ prescribed, in remembrance of the suffering which He endured on behalf of those who are purified in soul from all iniquity, in order that we may at the same time thank God for having created the world, with all things therein, for the sake of man, and for delivering us from the evil in which we were, and for utterly overthrowing principalities and powers by Him who suffered according to His will. Hence God speaks by the mouth of Malachi, one of the twelve [prophets], as I said before, about the sacrifices at that time presented by you: 'I have no pleasure in you, saith the Lord; and I will not accept your sacrifices at your hands: for, from the rising of the sun unto the going down of the same, My name has been glorified among the Gentiles, and in every place incense is offered to My name, and a pure offering: for My name is great among the Gentiles, saith the Lord: but ye profane it.' He then speaks of those Gentiles, namely us, who in every place offer sacrifices to Him, i.e., the bread of the Eucharist, and also the cup of the Eucharist, affirming both that we glorify His name, and that you profane it. ...
Accordingly, God, anticipating all the sacrifices3 which we offer through this name, and which Jesus the Christ enjoined us to offer, i.e., in the Eucharist of the bread and the cup, and which are presented by Christians in all places throughout the world, bears witness that they are well-pleasing to Him. But He utterly rejects those presented by you and by those priests of yours, saying, 'And I will not accept your sacrifices at your hands; for from the rising of the sun to its setting my name is glorified among the Gentiles (He says); but ye profane it. ...
Dialogue with Trypho [the Jew], Chapter 41, 117.
(Notice the tenses.)
The Hebrew is essentially equivalent to the Greek here, as to tense and intended meaning. That is to say, it's application is the same by a Jerome as it is for a Justin.4 The perfectness vs. imperfectness (Hebrew has no past or future tense properly speaking) is seen as irrelevant to time, but rather to prophetic certainty. (The ridiculousness of saying heathens offer a pure offering and fear Yahweh before Christ's time virtually demands that it be taken as a prophecy.)
The NJBC utterly flops when it asserts with a scholarly sobriety that there is in mind here "the pleasing oblatory practice (even without sacrificial animals) everywhere else in the world," that is, among heathens who sacrifices to devils, or at best, "other gods."5
The Soundness of the NJBC's Exegesis
It claims that מנחה (minḥa) is not used of animal sacrifice. It says:
It will be noted that none of these cultic terms has to do with animal sacrifice.
But this is just plainly wrong. In Genesis 4:4, a lamb offered by Abel is a minḥa unto the Lord. The word just means "gift" i.e. "gift [offering]." SImilarly in Exodus 29, lamb offerings are considered minḥath. The LXX's θυσία καθαρά a pre-Christian translation of the Hebrew minḥa t'hurah means "pure offering" and is in any case purely sacrificial language in this context.
1 Rabbi David Kimchi, A.k.a. RaDaK, Sefer Mikhlol, 45b (translation credit goes to @Der Übermensch. Cf. "Are there prophetic perfect tenses?"
2 Credit for the translation goes to @mevaqesh. I have absolutely zero approval from this person or the one above in citing using his translation.
3 Cf. Heb. 9:23. | "And this food is called among us Εὐχαριστία [the Eucharist], of which no one is allowed to partake but the man who believes that the things which we teach are true, and who has been washed with the washing that is for the remission of sins, and unto regeneration, and who is so living as Christ has enjoined. For not as common bread and common drink do we receive these; but in like manner as Jesus Christ our Saviour, having been made flesh by the Word of God, had both flesh and blood for our salvation, so likewise have we been taught that the food which is blessed by the prayer of His word, and from which our blood and flesh by transmutation are nourished, is the flesh and blood of that Jesus who was made flesh. For the apostles, in the memoirs composed by them, which are called Gospels, have thus delivered unto us what was enjoined upon them; that Jesus took bread, and when He had given thanks, said, “This do ye in remembrance of Me this is My body;” and that, after the same manner, having taken the cup and given thanks, He said, “This is My blood;” and gave it to them alone. Which the wicked devils have imitated in the mysteries of Mithras, commanding the same thing to be done. For, that bread and a cup of water are placed with certain incantations in the mystic rites of one who is being initiated, you either know or can learn." (First Apology, Chapter 66.) | Cf. 1 Cor 11:27-29.
4 Jerome is famous for 'introducing' the Hebrew form of the Old Testament as the 'original' into mainstream Christianity; before his time, the Bible of the Church was all in Greek (even if via the Old Latin in the West, for example, or Syriac in the East). He is famous for the phrase and concept of "veritas Hebraica" ("Hebrew truth").
5 Deut. 32:17.