Yes! I just got the game and I totally agree. I'm on my fourth generation and I swear there's only one horse in the village. It used to be really cool because when you invited a friend or partner to your sim 's house, they would often turn up on a horse and then tie him up with yours.
My solution for you is this, but it's not so great and doesn't make the extra horses that exciting but oh well. Make a family of two parents, get them to have a baby and then when the baby is grown up, adopt three random non related horses . Then, move out the offspring adult for you to play and thn you can visit your parent's house to see the horses. They will breed and other townies will adopt them etc.