This is the right way to do it. The liquid nails (or I'm betting probably polyurethane construction adhesive) is actually stronger than the wood itself. Smoothing it out w/ epoxy is a crazy extra step so I'm sure your cabinet is good for life. The curved, solid hardwood door is another, totally different method, there's a couple different ways to do that. One would be to use a bandsaw and cut the radius out of a thick piece of hardwood. The other is to use a mold and spray it w/ water while clamping until it 'memorizes' the bend, this takes a long time.
Sounds like you have a good cabinet maker
. IMO, using 2 or 3 layers of 1/4" birch or MDF ply (each layer would have a coat of wood glue brushed over to accept the next layer) would be the easiest way and would be as good as doing it that way.
I'll actually be making my first curved speaker cabinets pretty soon. I'm going to use 2 layers of 1/4" MDF, bent around the top, bottom, and middle brace and I'll veneer them w/ some mahogany veneer that I already have. They're TM bookshelf speakers. I'll definitely post the results once those are done.