It actually goes all the way down to County, Muni, and even School Board levels.
Again, it's a nuanced position that the Framers simply did not anticipate. On the one hand, they kept it so convicted people could still run for/hold office. The idea being some British loyalists couldn't still swoop in, have some show trial, throw a candidate for Congress in jail, and make sure they could serve the newly found nation after they got out. By the time Texas comes along and forms its Constitution, that wasn't a big concern anymore...so they just issued a blanket "you get convicted, you can't run for anything." Many western states have similar language in their constitutions because it wasn't a huge deal after the mid-1800's. I kinda see an in-between compromise. But no states are changing shit about this right now given the right stakes. Maybe in the 2032-33 cycles/sessions.
Texas could put a constitutional ballot measure to change this in the next two years. But considering the language of them typically reads like it was written after Trump got an online law degree and then drank a lot of cough syrup...I doubt it would move the needle much.