However... what would be the main point of distilliation?
This is easy to answer:
The dwarven work efficiency (not getting slow etc. from being sober) could be a result of amount of alcohol consumed. ACTUAL alcohol, as in taking proof into consideration.
Whereas not dying of thirst could be just any drink. Thus, if you give your dwarves small beer all the time, they'll not get enough alcohol to be fully efficient in the amount of time it takes to slake their thirst. Either they stay inefficient, or they have to drink extra units of booze to reach their alcohol threshold (which is also essentially just being inefficient since it takes time to drink more)
Whereas hard liquor could quench their thirst and make them maximally efficient until their next drink in one drink.
So distilling maximizes the efficiency of your non-brewer labor. You want that armorsmith to work as fast as possible? Then invest more of your less-valuable brewer time in distilling in exchange for making sure your more-valuable armorsmith is getting super efficient rocket fuel and not drinking too much or being too sober. Or do you not have any especially important dwarves? Then just go with beer, because it's not worth spending extra brewer labor just to save not-any-more-important farming labor, etc.
If subterranean biochemistry doesn't mesh with alcohol, then why are underground plants edible at all?
If underground plants aren't edible at all, why did Toady bother putting them in the game and allow dwarves to eat them for so long?
If underground plants are inedible, then how the hell did dwarves even survive without elves & humans?
Yeast making alcohol is not the same thing as mammals metabolizing fats and sugars. They are two completely different processes, and it's entirely possible that something could be chemically incapable of doing one but not the other. In fact, this is already the case with fats for example... we can gain energy from fats as animals, but you cannot make alcohol out of them. Same goes for proteins.
If it's convenient to the game design, why not claim that exactly that possibility is the case on the scale of the entire undergroud plant? I.e. they are entirely made of fats and proteins and [possible some unknown other chemical(s) that act similarly].
Your other two points follow from underground plants being inedible, which as I just explained, is not necessarily true just from them being unbrewable.