Do Texas BBQ joints have chicken? Goat? Lamb? Armadillo? Snake? Ostrich?
Or anything a bit unusual, like say "hamburger"?
Hamburgers are definitely not on a traditional Texas BBQ joint menu.

But yeah, yardbird (chicken) is common on the menu. So are turkey and pork loin. Pulled pork is becoming more common, most top places have it, these days. But that's definitely coming from the southeastern influence, when I was growing up, nobody served pulled pork. I'm glad they do now, though, it's tasty.
BBQ goat is more along the lines of something you'll find at a Mexican restaurant-- cabrito. It's delicious, but most BBQ joints don't sell it. Lamb might make its way on as a special at some places, occasionally.
Oh, my absolute favorite bite of BBQ, anywhere, is the smoked beef cheeks at local joint Leroy and Lewis. They're absolutely divine.
But in Texas, the "Holy Trinity" of BBQ are brisket, pork spare ribs, and smoked sausage. Every joint is judged primarily on those three. Beef ribs are also pretty common but they're huge and expensive and some places only do them on weekends and such.