It should be based on breeding, with rudimentary genetics, or some value assigned to horsed to calculate their quality on the racing field.

I do not want to have to personally tend or race the horses. A game from the manager point of view will suffice.

Bonus points if get to watch the horses race (but not actually having to ride them)

Look up games by Kairosoft, I am sure they have 2-3 games there which might be what you are looking for.