From experience, there's a bit of difference between a TB who was trained for steeplechasing, and one for flat racing. Flat horses are usually broken in before they turn 2 years old, tend to be 'sharper' as they're running over a shorter distance, and are more likely to have issues from being broken in so young (kissing spine, tendon and joint issues). Chasers (if bred for chasing, rather than flat horses who are kept on to go chasing) are usually left until 3 or 4 to be broken, so less issues with spine / back, and tend to be a bit more level headed (bred to go over 4 miles, so more stamina - sharp horses tend to not settle, so don't make the grade chasing) - also bred more with soundness in mind, as their careers can last much longer (flat horses as a rule only race as 2 and 3 year olds, then only the very good but not exceptional ones will go on another year or 2. The very best usually retire to stud, the less talented ones are sold / leave racing as 3 yo or go hurdling. Chasers will often continue to race until 10 - even into their teens). Chasers tend to have more bone, as the very fine flat horses wouldn't stand up to the rigours of jumping, but this isn't always the case. Injuries to chasers tends to be more tendons and ligaments. Chasers are, however, often stronger to ride, and as they're used to going miles at a gallop, you can't just 'wait until they get a bit tired' if they're getting a bit quick with you.
This is obviously a gross generalisation, there are exceptions to both.
Would I buy one for my 13 year old daughter? Hmm, it would really depend both on my daughter's riding ability, experience and temperament, and on the horse, and how well it's been reschooled. Some can be the very calmest, sweetest horses, easy going, easy to ride and trainable, where others can be very stubborn, quirky and real worriers.
That said, my 18 year old daughter has an ex-chaser who we had straight out of training. He was with a small trainer, who had him living out, so he's not as 'institutionalised' as some, and this trainer had given him a fairly varied life, including some show-jumping and schooling. He's not always easy, I've questioned my sanity a few times when he's done something particularly silly. He's not a great one in a stable (stressy) - especially if somewhere new, he can get quite quick and be hard to pull up, but he's a really lovely chap, tries really hard to please (although it takes a while for the penny to drop) and in just over a year she's managed to get him doing PC novice dressage and 90 SJ very nicely.