I think you've already received some of the best advice. I want to further the point of having a retirement plan for older horses. I also second the points of finding some good supplements (primarily joints) to support the older horse. It is also a really good idea (as mentioned) to do some research on feeding older active horses. There are many informative articles online that should guide you if you choose to take this path.
As long as the older horse is sound and in good condition, 19 years old is a good age for a beginner's first horse. I believe that temperament is important too as not all goldie oldies are bombproof or kid safe! There is a 25 year old warmblood mare that I wouldn't trust with kids!
When I was much younger, a very kind and honest 23 year old TB guided me over my first advanced courses. I look back and smile thinking of how that old boy loved to jump. With good care, and hopefully the horse you are interested in had good care up until this point in his life, you should have some rideable years on the 19 year old. Your daughter is young enough that she should get a good basis in riding and horse ownership from the old guy for some years before she wants an "upgrade". At that point, that is when the retirement plan comes into play.
I can't add anything new but I do support purchasing an older horse in the situation you gave. Good luck with finding a suitable price for him.