I agree with everyone, just look around for the horse that fits you the best, no matter what breed it is. Before I bought my horse, I was so sure that I wanted a quarter horse. I liked their stockiness, their dependability, their level minds... I looked at quite a few, and while I liked the way they looked, I wasn't falling in love with any of them. Then, one day, I found an ad for a gelding that sounded like the perfect horse, I just had to see him for myself. Two and a half years later, here we are! I couldn't be more happy with my fine boned, spirited, independent thinking thoroughbred.
On a side note, the horse that I took my beginner riding lessons on when I was 8 years old was a 14 year old Arabian gelding, and one of the best lesson horses at the barn I board at now is a 16 year old Polish Arab mare. I don't think you have to count them out just yet. Just look for older horses (10+) that are healthy and sound enough to jump. They are typically a hearty breed. As mentioned before, if they stay fit, they can perform for a long time.