I cannot find an option to ride bareback with level 10, and it is not specified in the level up notification, unsure where the info comes from.
I have however found a simple way to generate horses without a saddle: use the genetics tool to create a foal from an existing horse, then age them to adult in the same window, more often than not they grow up without a saddle. I have had no luck for the bridle, it appears you have to have one.
I've tested it in live mode, the sims do ride the horse in the riding outfit they were generated with, with no saddle.
