From what I've witnessed, dogs and cats need annual exams for shot and preventive meds and dental care, etc. but domestic exotics animals like birds, bearded dragons, tortoises, snakes, etc. don't typically need a great deal of veterinary care unless something major goes wrong. I can't see where you would be able to be solely an Exotic Vet unless you were just wanting to have a part-time clinic or maybe a mobile Vet clinic (which I think would be cool).
That being said, Dr. Theilen, also known as Dr. T from National Geographics "Lone Star Vet" show and previously on National Geographics "Dr. K's Exotic Animal ER" , recently opened up an Exotic Vet Clinic with Dr. Hoppes (A Veterinary Professor at Texas A&M), in Grapevine Texas. They seem to be successful but they also both have a great deal of credentials behind their names and Dallas/Ft. Worth is a really big area.
I live in Houston and have exotic animals. I'm also a zookeeper so I take my exotics where we take our zoo animals, which is Texas A&M school's vet clinic. Prior to that, I had a hard time finding vets in our area that specialized in Exotics. They treat them, but really don't know a lot about them. When I search Exotic Vets in Houston, I get vets that treat both domestic (dogs & cats) and exotics. This again tells me that it's hard to maintain enough clients to support just an Exotic vet clinic. But it really depends on your area.
What you might consider is being a Vet at a zoo. But I can tell you that the zoos take the best of the best when they hire their Vets, and they don't typically have but one, maybe two, on staff.