Exactly, because there was no right to horse ownership in the Bill of Rights.
There are listed no 'rights to ownership' to most items in the bill of rights. It was just assumed by the founders that one would, of course, have a right to own their transportation like they would a fork to eat their food or a chamberpot.
The Bill of Rights didn't grant rights, it protected very specific rights from government interference. I doubt the founders would have ever thought of a day when other items like basic transport, would be up for government regulation, otherwise we would have a Bill of Rights protecting those things as large as Jefferson's library.
What the Bill of Rights protected were specific rights they saw being stripped by powers for the specific reason of suppressing the individual and liberty. The British, and other repressive regimes, didn't come into homes regulating chamberpots or what fork you used, but they did strip people of the right to defend themselves and mandated beliefs. They quartered their soldiers in people's homes and imprisoned people without any due process.
The importance of the very specific items called out in the Bill of Rights all have a common theme- the citizen is sovereign.