Is the commercial tub-and-tile epoxy paint really that
bad?

In a word, yes. I used to rent an apartment in which the bathroom sink
had been re-done in that stuff, and it was fine for the first month or
so. Then it started chipping/flaking/peeling up, and it kept doing
that for the whole year and a half I lived there. Don't bother. I
think it would be even worse if you tried to build that stuff up to any
thickness.
If it were my sink, I'd probably try to make a new drain that comes in
from underneath and fits the hole as close as possible, and seal around
that new drain with silicone caulk, realizing that it wouldn't be a
really permanent or durable solution.
Good luck,
Andy