Why does God punish us for our will?

The definition of free will that I and, I would venture to say, most other Christians use is what is called libertarian free will. It can be summed up as the ability to do otherwise, but it means that for any action you take or decision you make, nothing caused you determinitively to make that decision. This includes your environment, people influencing you, even your own desires and urges. You are the ultimate origin of your decision.

Our desires can and do influence us, but they do not rule us, and we can choose to do what we do not desire.

What you describe is almost more like compatibilistic free will, which I would agree is no free will at all, rendering our decisions a mere necessary byproduct of our circumstances, and hardly something it would be just to punish us for.

Don't worry about the immensity. Christian theology has grappled with "free will" for ages, and after fruitful engagement with modern philosophy, Christian philosophy has produced a logical and realistic account of the phenomena of "free will" in human souls.

By the way, the most likely interpretation of the quote (as explained in this Philosophy.SE question) is

"you are free to do whatever you desire, but you are not free to choose your desires"

Christian theology concurs with the above by appealing to empirical data that it's very easy to assent to follow our desires, good or bad, but especially bad ones. Christian theology teaches that our free will is BROKEN, biased to do what we know to be wrong. Most groups agree that since we are born our will is dominated by concupiscence which hijacks our attempts to do good. Paul talks about this in Romans 7:14-25. No one will dispute the fact that all humans are "enslaved" by self-centered desires ("not free"), unless they exert extra effort to overcome them. Sometimes natural love makes it easier to wish the good for others (children, lovers, etc.) and then to actually do the good acts, but we all realize it's not sufficient sometimes.

But please understand punishment correctly. In Christian theology, God punish us for the sinful ACT (fruit of the will), but NOT usually for the WILL per se, especially if we are fighting our own desires which induce us to will harm. Even if we succumb to sinful acts or to "sins of the heart" (like cursing others, lust, or envy), the punishment is mitigated by many factors that are "straitjacketing" the will such as addiction, ignorance, biological factor, passion of the moment, history of abuse, etc. People of goodwill recognize that they did wrong (the Catholic formula is "in my thoughts and in my words, in what I have done and in what I have failed to do") and would be induced in their conscience to confess their sins in front of God and to cry out for deliverance of their broken will. God (out of his abundant mercy) will forgive the sins and (upon request) will infuse grace to strengthen their will so they can potentially succeed when they try again.

Some Christian philosophers such as Eleonore Stump proposes that the will has three positions, not two: assent, rejection, or quiescence. Even though our will is broken, we can at least opt for "quiescence", the posture that allows God to strengthen our will by grace (i.e. increasing our love). This is God's way of preserving our free will; God will not force his grace on us. See this paper Grace and Freedom: Examining Stump's View of the Quiescent Will.

CONCLUSION

Christianity teaches that we cannot "assent" to heal our own broken will ourselves since we are trapped in the conflict of wanting to love and don't want to love at the same time. We need God's supernatural assistance which God offers unilaterally but which we are powerless to positively "assent" to receive it (because of this conflict).

But by freely putting our will in "quiescence" setting instead, we can acquiesce ("cease resisting") to receive the needed grace to fix our broken will. In this video, Eleonore Stump likens it to a child agreeing to receive a vaccine injection where the child neither rejects nor positively desire the vaccine (how many child says "please give me a vaccine?"). God only punishes us when our will is in the "rejection" setting (i.e. refusing the gospel when it is fully understood), thus refusing God's grace that could have assisted us in wanting to want to love (second order will) so we are put in the process of growing to want to love and be loved (first order will).

OP: What we want at any given moment is determined by who we are and our environment

True. But, if one is religious, one objectively knows whether something is wrong, and can choose not to do it for that reason, suppressing what one happens to want at that moment.

And if one is a Christian, it is a duty to develop and grow one's character to no longer want to make choices that are known to be wrong. Eventually there won't be any inappropriate wants.

That is, we get to decide "who we are" by changing ourselves. That's what the word "repent" means:

To repent is to regret so deeply as to change the mind or course of conduct in consequence and develop new mental and spiritual habits.
— repent | Online Etymology Dictionary

The trick is to regret it before it happens.

The Basic Argument seems to be based on there being no objective morality, which for Christians doesn't apply.

Christians don't avoid making bad choices for fear of punishment.
Christians make good choices because that's the right thing to do, and eventually because what they want to do ends up being the right thing.

(I also disagree with the idea of God "punishing" someone, as opposed to someone suffering the consequences of their choices. But that should be a different question.)

Free will--in the sense most Christians believe in it--is misrepresented by Schopenhauer's argument; I will summarize 3 reasons for rejecting this characterization of free will:

1. It incorrectly conflates our "free will" with our "desires". "Free will" does not describe "freely wanting things" - this is just a sleight of hand using different definitions of the word "will". Free will refers to the non-deterministic ability to choose among multiple options.

2. It is possible to want 2 mutually exclusive things at the same time. A person on a low-sugar diet may genuinely want the promised results of the diet and want a nonstop menu of sugary snacks. In such circumstances, the decision made is not simply the aggregation of various desires, it is a prioritization among desires. Many of our choices pit short-term gain vs. long-term loss & vice-versa.

3. It is possible to change one's desires--this is a major plank of Christian belief

22 But the fruit of the Spirit is love, joy, peace, longsuffering, gentleness, goodness, faith,

23 Meekness, temperance: against such there is no law.

24 And they that are Christ’s have crucified the flesh with the affections and lusts.

25 If we live in the Spirit, let us also walk in the Spirit. (Galatians 5:22-25, compare the alternatives discussed in vss. 16-21)

If we invite that Spirit into our lives it will begin to change our desires.

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Conclusion

We may desire many things, some of which conflict with each other. Free will gives us the capacity to act, to choose which voice we will yield to.

I don't necessarily contend with the view that humans pick the desire that is the most compelling in the moment, but I suggest the analogy would be better cast in different words:

We choose which voices to listen to; we choose which influences to entertain. Many voices compete for our attention; we control which speakers get turned up and which speakers get turned down. While we may well follow the voice that speaks the loudest, we set the volume.