Don't most animal have to be in heat to cause trouble?
The heat cycle for a mare is about 5 days every 3 weeks. Not just stallions but geldings or more rarely other mares will react to them when they're in heat, which can be a pain, even if that particular mare isn't too susceptible to mood swings. Back when horses were often tied side by side, turned out together in groups, etc., a nice, stolid gelding would have been preferable for most.
Stallions are always in heat. IMO testosterone is the most powerful substance in the world. I sincerely doubt any sensible person who used a horse for ordinary transportation would have considered a stallion as an everyday mount.
The OP's premise is that it's sad when research shows you can't have something work the way you want in a story, which is different from not doing research at all - something that spoils a lot of stories in all genres.