Goodness me, there are a lot sweeping statements on this post.
NH horses are a pain to teach to jump and dangerous xc? As with all horses depends on the horse. I have had a Grade 1 winning hurdler who was a brilliant show jumper, very accurate and never had a fence down from the moment we started re trading him. He was safe xc but not over bold. I have had a slow but sweet 3 mile chaser who I would take down to any hedge but made for,the gate if I saw a rail coming unless it was over 3'6".
I have had sprint bred horses who have jumped and ones who haven't, ones who tried but never got it etc.
TB's are just like any other horse, they are all individuals. I would suggest you buy on good conformation and temperament if you can find out a bit in advance. After that, you may have found a brilliant one or it might be not what you want at all, buying horses straight out of training is a lottery.