I think a lot of the debate in this thread comes from a misunderstanding of the various types of "stock" trailer, and I suspect some folks have never seen the old type of cattle trailer that SR and others are objecting to.
I would not haul a horse, particularly a TB straight off the track, in an old fashioned *cattle* trailer. Lots and lots of *combo* trailers, that are called "stock" trailers, are absolutely fine, and yes, the difference is the headroom.
I hauled for years in a standard 16' stock designed for both horses and cattle, and it hauled like a dream. Like the previous poster, I found it much easier to teach horses to load in one and many horses who were bad haulers in a two horse hauled fine in it.
My brother-in-law who runs a beef cattle business has a "stock" trailer that you'd be happy to haul a horse in. I don't see many of the old style, short cattle trailers any longer, most folks have gone to the taller, combo type as they are much more versatile.
To the OP - I'd be willing to bet money that a professional transport person has the all-purpose type stock trailer that is fine, even preferred, for horses. Interestingly enough, it was professional haulers who first understood how pratical and useful stock trailers were and popularized their use. Back in the day when I was a junior, everyone hauled in the big Frank Imperatore vans or two horse bumper pull Hartmans. Then stock trailers were popular, and now slant loads.