I remember when I was in fifth grade my mom bought me an 18 year old pinto gelding. Everybody laughed at her for spending 750. On an "old" horse. I rode that horse until I graduated high school and left for the navy.
My first horse show I remember doing really well, but not getting the top ribbons. My mom went and asked the judge what I could improve on, and she said I wasn't riding him to his potential - turns out he had been a campaigner on the show circuit with Register of Merits in most classes from halter to western to jumping - the judge was shocked to find out mom had got him at the local meat auction. I guess someone thought he was too old and dumped him.
I learned so much with him, won tons of classes, and he was the most kind and gentle horse a kid (or parent) could hope for. He died at 38. Rest in Peace, Hey Sugarfoote!!!
My own oldest daughter learned how to ride on a 37 year old named Bar Suds.
If the horse has been cared for and looks good, I would not hesitate to buy a 20 year old. In fact, that is about the age I will be looking for next year for my now 6 year old daughter's first horse.