really depends on the particular TB, honestly if a horse used to have big bucking problems, etc but has been doing better it still may be a bad choice as horses who learn they can get away with something can regress rather quickly in their training and behavior. I would go with something that is a proven steady eddy, been there done that kind of horse, even if its not fancy or anything (just my personal opinion)
I have a TB and I would not put a beginner on her, she is 16 yo but is too sensitive and if she knows you dont know what youre doing or are not paying attention she will take advantage. My Saddlebredx who people tend to assume will be a bit of a nut when they hear he is half saddlebred is so good for beginners though bc he would really rather go as slow as possible :p so, proof that it really just depends on the particular pony!
good luck in your horse search!!