Well, since the thread said endurance and
competitive trail, I can speak for the competitive trail side.
Competitive trail is a broadish term. When I say it, I mean a pace race where you have a window of time and ride between 15 and 40 miles a day for two or three days. I particularly do rides sanctioned by NATRC, so that's where most of my experience is coming from.
horse can do Novice/CP level rides if you put a little effort into conditioning. These rides are usually around 20 miles a day for two days and are usually paced at around 3.5-4.0 miles per hour. For most horses, that means fast walking and a little trotting. I've seen huge drafts and little ponies complete and even win. I see a little bit of everything
Many horses can do Open level rides. These are around 30 miles a day (sometimes 35/40 the first day and a shorter day after) at a 5 or 6 mile per hour pace. Doesn't seem like a huge increase in pace, but it leaves you trotting where ever you can and cantering when it's good, especially when rougher terrain has made you walk. They also put you down harder trails than Novice/CP sometimes. Also, a lot of horses can complete one Open ride or two... But don't last the whole season.
Once you get Open, it's a lot of Arabs. Arabs do very well. We also have a lot of fox trotters, TWH, half arabs, morgans, lean QHs, and an odd appy. I've noticed it isn't so much breed as body type. Lean, lithe horses; built like greyhounds. A pretty fit bunch.
CTR is also a strategy game. You can beat a fitter horse if you plan better. Find a good place in the pack, a good way to make pace time (some horses do better with long trots and some with intermittent smaller ones), decide where to stop and how long, etc. Plan your ride so you'll be a couple minutes a head about two miles out from the vet check and walk in. Your less fit horse will pulse down better by walking in than a fitter horse will do by running in, if that makes sense.
So yeah. You can make most horses into CTR horses if you know your stuff.