When I was looking at lesson barns in New England a few years ago there were many many places with a posted weight limit right on the web site, sometimes as low as 175. Given that the usual lesson student is a teenager, it makes some amount of sense to keep horses appropriate for that, and it mostly seemed to be the big places with very busy lesson programs that had a flat advertised limit. In any case it wouldn’t surprise me at all to see same.
It cuts both ways, though. I’ve talked about it with a barn manager who runs a busy lesson program, and she says that she has horses that can take even beginners over 300 lbs if they’ve tolerably balanced but that much lighter-but-bouncier riders have been a problem, and so having a flat limit posted would be pretty useless.
My personal experience is that if one is a serious (and hopefully somewhat competent) student it’s just about always determined on a case-by-case basis, and as a regular student on the tall-and-heavy side that it’s been a concern but not a fatal one. Of the two trainers I ride with regularly, I lease a horse from one who seems happier to have me on him than some much lighter riders, and use a lesson horse of the other but keep it to low-impact work. (Jumping BN and below, basically.)
There are a whole lot of horses in the world. I expect there’s one for almost anybody.
PS: I was also recently in Aruba, which has some spectacular trail riding, and there are at least six places that offer same and they all have weight limits posted someplace as well as some height limits. In actually contacting them by email, though, they were a lot more flexible by prearrangement. (I was expecting to end up riding one of the smaller population of quarter horses on the island, but I ended up on the biggest Paso Fino I’ve ever seen! Also, the Natural Pool ride via the peak of Mt. Arikok is of most epic awesomeness. Even in tropical heat!)