You probably have a very good balsamic, something probably not in most grocery stores.
I think most people don’t realize that a great balsamic isn’t the acidic, tart, watery stuff, but something more viscous and damn delicious all on its own (though acidic ones have their place in cooking, like sauerbraten).
I have a few, one that’s just for salad or as a finish, others I’d use while cooking.
You lose as much alcohol as you do water, as they’re chemically bound together.
Alcohol doesn’t really cook out of a dish.