The author of Hebrews is not identifying Melchizedek as a pre-incarnate appearance of Jesus or what have you, but as a type or pattern for Jesus' priesthood.
Argument From Chronology
The first thing to note is that this passage is a reflection on the Melchizedek narrative in Genesis through the lens of Psalm 110, which was introduced back in 5:6 demonstrating that Jesus did not take upon himself the office of high priest, but was appointed by God who declared to him, "You are a priest forever, in the order of Melchizedek."
The author of Hebrews notes that the psalmist's oracle establishes the need for a greater priesthood than that of the Levitical priesthood in the order of Aaron. If the that priesthood had been adequate, he asks in verse 11, why would there be an oracle written some time later declaring someone else now as priest forever and of a different order?
In other words, by the author's own interpretation of Psalm 110, the priesthood being established there is something that comes after the Levitical priesthood and therefore demonstrates its inadequacy to meet the needs of the people. We cannot make sense of the author's argument, however, if he believed that Melchizedek was Jesus, since Melchizedek was priest not after the Levitical priesthood, but before.
Similarly, we see that the author of Hebrews considered it important that the oath which made Jesus a priest (5:6) came not before the law, but after (7:28). Melchizedek's priesthood obviously came before the law; therefore it cannot be the same person.
Analogy vs Identity
As you note in your question, there are a number of analogies/similes in Hebrews 7 comparing Melchizedek and Jesus. The two key verses here are 3, 11 and 15. I'll return to verse 3 in a moment, but the analogies of verses 11 and 15 build on the first point. In verse 11, the author, reflecting on Psalm 110, notes that it looked forward to another priest who was to come. In 15, what has been said becomes more plain because another priest has appeared - one like Melchizedek: namely Jesus.
In other words, the author of Hebrews, interpreting Psalm 110, notes that it looked forward to another priest who would be like Melchizedek. He does not say that the psalmist looked forward to the return of Melchizedek or some such thing.
Returning to verse 3, there is one more crucial point - that Melchizedek is made subordinate as a type to the Son of God and not the other way around. Yes, the author of Hebrews says that Jesus was like Melchizedek. But in verse 3, he is careful to note that the resemblance is because Melchizedek was "made like unto the Son of God." From this it is clear that the author understands God's intention that this historical person, Melchizedek, and his story should be written in such a way as to resemble the Son of God - i.e. the silence of the Genesis narrative on Melchizedek's genealogy, paternity, maternity, birth, death, etc... was deliberate foreshadowing of an eternal priesthood based not on laws regarding ancestry, but based on the power of Jesus' indestructible life. As David Allen puts it (NAC), "It is this phrase, 'like the Son of God,' that the author uses to indicate two important truths: the greatness of Melchizedek; yet he only resembles someone greater."
This is reflected in the final phrase of verse 3 as well. While when quoting the psalmist in verse 17 and 21, as well as when describing the priesthood of Jesus himself in verses 24 and 28, the author uses the phrase εἰς τὸν αἰῶνα, here at the end of verse 3 we see that he instead is careful to use a different phrase, εἰς τὸ διηνεκές, translated in the KJV as "continually." BDAG defines this phrase so:
pert. to being continuous, without interruption, always of time εἰς τὸ δ. for all time, without interruption ... μένει ἱερεὺς εἰς τὸ διηνεκές remains a priest for all time (i.e. Melchizedek’s priesthood goes on without lapse) Hb 7:3.
Arndt, W., Danker, F. W., & Bauer, W. (2000).A Greek-English lexicon of the New Testament and other early Christian literature. Chicago: University of Chicago Press. (emphasis original)
If the author wanted his readers to understand that Melchizedek was Jesus and that his priesthood was truly eternal, we would almost certainly have used εἰς τὸν αἰῶνα in order to reflect the psalmist's prophetic word as he does when describing the nature of Jesus' priesthood. However, he uses a different word to describe the nature of Melchizedek's priesthood. This is because the author did not consider Melchizedek to be an eternal and indestructible being, but the shadow of one, typological shadows being a major theme of this section of the book (cf. 8:5, 10:1).
In reply to your objection concerning the unchangeable order...
It is not Melchizedek's priesthood or order that is unchangeable in verse 24, but Jesus' priesthood on the basis of his indestructible life. While Melchizedek's death is not recorded in Genesis so as to make him a type, he of course did eventually die as a historical person as far as the author would have been concerned. With him would have died his priesthood.
But the author draws the contrast that Jesus, having been raised to eternal life beyond the possibility of death attains to a priesthood that can likewise never perish - especially since it had been sealed with an oath. Melchizedek was not raised to such a life, nor was such an oath given to him; and his priesthood was not forever.