IANAL, (nor am I the OP,) but my understanding is:

You can do X for no reason != You can do X for any reason

i.e. they can fire you when they wish to, but if you can demonstrate that the reason for firing was illegal, then it the firing is not for "no reason", but for an "illegal reason."

Can't they just fire the alcoholic because they don't like his tie, then? Or, say, assign him an impossible project and then fire him for not completing it?

They can. At will means they don't have to provide a reason. It gets complex though when the fired employee claims he was actually fired for illegal reasons. E.g. they might claim they were fired because of their age, because they were a woman, etc. Since most companies like to avoid lawsuits (but win the lawsuits if they do happen) they typically have internal HR processes that make it much harder to fire people. These require managers to document everything so that when person X claims they were fired because of their age, the company can hit back with "no, actually we have this whole file documenting all the instances in which you screwed up".

> Or, say, assign him an impossible project and then fire him for not completing it?

What you're trying is in UK parlance called "constructive dismissal" and it's not good; many jurisdictions make this risky.

https://en.wikipedia.org/wiki/Constructive_dismissal