I personally don't see any problems with this. For example, in most cases we can agree if someone had free will or no when they committed a crime. Even if I prefer compatibilist view and you are not, we'll agree in the most cases. In the cases when we may not agree the reasons of disagreement will not stem from this fundamental disagreements, but from things like should we treat state of affect as a state when a human has no free will.

So, a compatibilist view is not incompatible with the world we live in, but moreover, it is needed to keep our world functioning. The world we live in is mostly artificially constructed. Welfare and justice systems are not "genuine", they are artificial constructs. They play a role in our society and the ideas of "free will" and "guilt" are constructed also, and they are tweaked to make our systems to work better. If you assume that free will and guilt are "genuine" or God given, then you can't tune them to better match their purposes. You are losing agency this way, losing part of your free will, you can't consciously and reasonably discuss if state of affect should be an exception from the rule "any person has a free will". You'll be forced either to skip the discussion, or to resort to some kind of theological arguments.

But if you accept, that "free will" is a social construct, then you can easily identify the affected variables: it is all about punishment for crimes or awarding people for their pro-social deeds. You can think of how "state of affect inhibits free will" can influence all these goals, you can think of the possibility of people simulating state of affect (or even nurturing their personal traits that increase the probability of entering state of affect) to avoid a punishment. You can think rationally, logically and to pick a solution that benefits the society the best. Those very society with baked in idea of free will. Or you can choose to believe "free will" is God given, because of an irrelevant linguistic argument, and lose the ability to make our world better.

> most people act in ways that clearly signal a non-compatibilist view of free will.

Of course, we are not living in quantum mechanics we live in a world that is constructed by people. I mean, all this is built on top of QM, but QM laws do not manifest themselves directly for us. We have other explanatory structures to deal with everyday physics. But even physics doesn't matter that much: I turn the switch and voila I have light, and the heck with conservation of energy. I can talk to you, despite we are residing on different continents, 1000s of km don't matter. If I want to eat I do not try to kill some animal to eat it nor do I gather seeds and roots in a wild to eat them. I go to work and do something, get my salary and buy food in a local store. We are living in an artificial world with artificial rules. Free will is part of this world. Of course we talk about it like it exists. We talk about it like it is a universal truth. Relativity I mentioned above doesn't show itself most of the time, because the world is constructed in a way, when we can agree about someone having it. Situations when this is not the case are very strange and can be even punished: manipulation (which is come close to taking people's agency away from them) is deemed amoral.

The world constructed so we can ignore that free will is just an illusion, moreover it is constructed to think about it in terms of free will, so you'll have issues thinking about it in other terms. Like you'll have a lot of issues trying to calculate aerodynamic of a plane relying on equations of quantum mechanics.