C.W.
My son (Aspergers) went to an extremely limited diet at around age 3/4. You're right, there's no such thing as "They'll just eat when they're hungry" with these kids.
We worked painfully slowly. For a while, we let him stay within his comfort zone. Then we started requiring "one tastes", which often resulted in gagging, if not vomiting, and we adjusted to "one licks" until he was a little more mature to reason with.
We also taught him to hide his own food. Instead of hiding his "no" foods on him ourselves, we showed him how mashed potatoes could help a few peas go down, or how cheese on a bite of chicken would change the whole mouth feel.
As he got older, we traded bites of "no" foods for bites of desert. An entire small hamburger would earn an entire serving of ice cream. Smaller increments would mean less ice cream.
Pediasure was our friend. Yes, we brought our own food or fed him before/after events outside of the house.
He's still a picky eater (turning 13 this month), but NOTHING like how he used to be. He now eats almost all meats (vs only processed chicken nuggets), miniscule portions of vegetables (vs none), and will at least TRY most new foods, whereas the thought used to make him physically ill. What he does eat, he eats like a horse.
He's still very, very skinny, but has hit 5'7 and a size 10 shoe before becoming a teenager. His doctor is content with his growth.
ETA: Our occupational therapist had no suggestions on top of what we were already doing, nor did our developmental pediatrician. They were, however, both aware of the situation.