Seeing as there is only one mediator between God and man, what’s the role of the prophets?

To summarize, the role of the prophets was to deliver messages from God, generally in the form of warnings along with the future consequences of failing to listen and obey God's warning.

In one way they are mediating a message between the population and God but they were not THE Mediator between God and man in the same way Jesus is. How did Jesus mediate between man and God? By removing the gap between God and man caused by our sins (for those who accept the mediation).

“Seeing as there is only one mediator between God and man, what’s the role of the prophets?”

The mediatorship of Christ, which is absolutely unique, is not exclusive to all other mediatorship, in the context of the passage, only exclusive of other Reconcilers with God in the ultimate sense.

1 Timothy 2:1-6 (DRB) I desire therefore, first of all, that supplications, prayers, intercessions, and thanksgivings be made for all men: 2 For kings, and for all that are in high station: that we may lead a quiet and a peaceable life in all piety and chastity. 3 For this is good and acceptable in the sight of God our Saviour, 4 Who will have all men to be saved, and to come to the knowledge of the truth. 5 For there is one God, and one mediator of God and men, the man Christ Jesus: 6 Who gave himself a redemption for all, a testimony in due times.

Rather than His Mediatorship (albeit unique by nature, and exclusive in this sense alone) precluding any mediatorship of others (e.g. Job 42:8; Rev. 5:8; 8:4; etc.), this passage uses the mediatorship of Christ as the basis upon which intercession of others, who are not Christ, is made meritorious or effective. Contrary to any notion that all intercession or mediation is rejected here, it teaches that participiation in (not usurpation of) Christ's mediatorship is the will of God: "For this [intercessory prayer] is good and acceptable in the sight of God our Saviour, who will have all men to be saved, and to come to the knowledge of the truth." It would be a contradiction in terms, were we to say Christ's mediatorship excludes all other mediatorship just after saying we should intercede on behalf of others—that misinterpretation must give place to the prior affirmation of other forms of mediatorship in verse 1.

What is meant, then, is that Christ, being the incarnate Word, is the ultimate Source of reconciliation for mankind—"the mediator between God and man." The Mediator of the Covenant of Redemption (Heb. 9:15). What it cannot mean, due to the many instances of approved mediation (including one glaring example, the priesthood) is that his mediatorship precludes others' mediation in other ways.

It is in this sense that exclusivity is denoted by "one" (as in "one" mediator between God and man).

Another way of looking at it might be that priests themselves (one might say, the highest form of mediatorship outside that of Christ's) themselves are in need of reconciliation with God, whereas Christ is the Reconciler of all in Himself, being God and man Himself, uniting God and humanity in the ultimate sense in His own person. Priests in a sense mediate salvation, but so do prayers for conversion in a way also by laypersons, but clearly this is quite different from God reconciling man to Himself in the person of Jesus Christ, true God and true man.

“To me it seems clear that all the promises of God are in legal force for all who are in Christ, but it’s not because they were spoken by the prophets, but rather because they were spoken by God. The prophets were merely listening in and relating what they heard.”

This is quite true in a sense, but authorities carry authority in their own right, something Jesus teaches quite clearly. They have authority from God, to be sure, but nonetheless they possess this real authority with which they have some freedom to exercise it as they will.

Matthew 23:1-4 (DRB) Then Jesus spoke to the multitudes and to his disciples, 2 Saying: The scribes and the Pharisees have sitten on the chair of Moses. 3 All things therefore whatsoever they shall say to you, observe and do: but according to their works do ye not; for they say, and do not. 4 For they bind heavy and insupportable burdens, and lay them on men's shoulders; but with a finger of their own they will not move them.

I don't know if this authority comes quite reaches the level of "whatever you bind on earth will be bound in heaven" (Mt. 16:18; 18:18), or "whose sins you forgive they are forgiven them; whose sins you shall retain, they are retained" (Jn. 20:21-23) the authority given to the Apostles, but it is quite able to be used at the discretion of those who wield it—even, as can be seen in Jesus' example, to bind what are unfair or heavy laws (lowercase 'L') on subjects to said laws, and even if they are, hypocritically, not obeying it themselves.

“Yet God says that He does nothing without the prophets. That sounds more like a choice by God than a legal requirement that we can’t talk to God without the prophets.”

The fundamental reason there are prophets and messengers is simply that "no man can see God and live" (Ex. 33:20; cf. Jn. 1:18). Even God appearing in a visible form (Isa. 6:1; cf. Jn. 12:41) would be an "angel" or "envoy" (מלאך) of His own presence (Isa. 63:9), you might say.

“Being “in the image of” seems to be related to the biblical idea of human authority on the earth, but it seems to indicate far more than just decision making authority. It goes toward the idea that we should have a resemblance of God and people that know Him should say that we remind them of Him.”

Indeed, if God is willing to save humans it must certainly be that the image of God which is in them by nature represents something of His own nature, and as such is of a very high, quasi-infinite worth. How can else can we explain God finding need to give mercy to people who all hate Him? There must be something wort restoring that He saw in them, something much greater and more profound than simply 'given the decision making authority on earth.' "In the image of God He created them" (Gn. 1:27) is essentially saying 'God put something of Himself in them.' I think this is where man's great authority comes from and cannot really be separated from meaningfully.

Being made in the image of God (Latin: imago dei) refers to the immaterial part of humanity. It sets human beings apart from the animal world, fits them for the dominion God intended them to have over the earth (Genesis 1:28), and enables them to commune with their Maker. It is a likeness mentally, morally, and socially. Today, we still bear the image of God (James 3:9), but we also bear the scars of sin. Mentally, morally, socially, and physically, we show the effects of sin. More information here: https://www.gotquestions.org/image-of-God.html

Prophets (men) were tasked with speaking God’s Word to people. In the Old Testament, this included both proclaiming God’s truth to others and revealing God’s plans for the future. Some of the prophets also performed miracles and healings.

Jesus, however, was no mere mortal. Jesus, though born of a woman, was God incarnate and was without sin. That is why Jesus is the ONLY mediator between God and men because all men are sinners.

Jesus is the Word of God (John 1:1). He does not simply speak the Word of God as a mere human prophet, but is Himself the Word made flesh (John 1:14). He is the final word, the ultimate revelation of God: “In the past God spoke to our ancestors through the prophets at many times and in various ways, but in these last days he has spoken to us by his Son, whom he appointed heir of all things, and through whom also he made the universe” (Hebrews 1:1–2).

Old Testament priests served as mediators between humans and God. It was the priests who offered sacrifices on behalf of the people. Jesus is our Mediator and our High Priest: “For there is one God and one mediator between God and mankind, the man Christ Jesus” (1 Timothy 2:5).

Hebrews 4–10 details how Jesus is our ultimate High Priest and how His priesthood is far superior to the Levitical priesthood of the Old Testament. The writer of Hebrews also explains how the Old Testament system of priests served to foreshadow the ministry of Jesus. The Levitical priesthood of Aaron’s line was not intended to continue forever. Jesus’ priesthood is eternal.

Hebrews 4:14–16 says, “Therefore, since we have a great high priest who has ascended into heaven, Jesus the Son of God, let us hold firmly to the faith we profess. For we do not have a high priest who is unable to empathize with our weaknesses, but we have one who has been tempted in every way, just as we are—yet he did not sin. Let us then approach God’s throne of grace with confidence, so that we may receive mercy and find grace to help us in our time of need.” With Jesus as our High Priest, we can go before God boldly, knowing that Jesus has true compassion on us and that, through Him, we will experience the grace and mercy of God (see also Hebrews 10:19–23).

Hebrews 7 shows how Jesus is a priest after the order of Melchizedek. Melchizedek was both a priest and the “king of Salem” who blessed Abraham (Hebrews 7:2; Genesis 14:18). Likewise, Jesus is not just a “priest forever,” but also a king. Source: https://www.gotquestions.org/Jesus-prophet-priest-king.html

Prophets prophesy. Jesus mediates. ONLY Jesus can mediate between us and God.

The difference between a prophet and a mediator is actually quite simple.

• A prophet's job was to deliver messages FROM God to humans; see Deut 18:22, 2 Kings 5:8, John 12:38, Jer 23:33, 46:13, 51:59, etc.
• By contrast, the mediator's function was to convey messages from humans TO God, something like the priest in OT times did. However, the over-all function of a mediator was to bring two warring parties together, usually by brokering an agreement (or covenant in the language of the Bible) Gal 3:19, 20, 1 Tim 2:5, Heb 8:6, 9:15, 12:24, Isa 42:6, 49:8, Job 9:33, Deut 5:5, etc.