For the most part, no you would not be missing out by just playing one class. The devs specifically designed the game so that anyone can play their dream class and nothing else if desired.

The only think you'd be missing (aside from faster queue times) is whatever story is tied to other classes and roles (the latter starting with Shadowbringers). There is a short storyline in Shadowbringers that requires that you complete every role questline to access, but it's hardly mandatory.