Fox hunters go everywhere trail riders do, and just as fast or faster. And staff can spend a lot of time bushwhacking.
Actually, I think hunting is more “dangerous” in terms of why you’d wear boots than trail riding is. Riding in the hunt field is a lot more disciplined than riding in a recreational group, meaning that you “suck it up, soldier” and ride when/where/how fast the rest of the group is riding. And, as I mentioned, staff end up having to go literally anywhere as necessary, often quite fast, in order to do their jobs. In the hunt field, you frequently don’t have the luxury of slowing down for hazardous conditions.
As wateryglen stated, boots aren’t common in hunting. Most people feel they cause more problems than they fix. As mentioned, anything with rubber, neoprene, etc. will hold heat on the leg. Any boot can get bits of ‘stuff’ down inside that chafes, whether it’s vegetation, silt from water crossings, mud, etc. Improperly fitted or applied boots or bandages, or ones that’ve shifted, can damage tendons. Any boot can catch on things. Etc., etc., etc. Lots of downsides for a very questionable level of protection.
The lone exception seems to be front bell or over-reach boots for horses prone to stepping on their own heels. But those are strictly as necessary, and I’ve only seen plain, soft rubber that is unlikely to rub or pick up debris.
ETA, to address the “why use boots” issue. Horses are rarely seriously ‘attacked’ by sticks, brambles, etc. and–IMO–that’s a really stupid reason to use boots. The front of the leg is bone and skin, like your shin. A trail-ridden horse may bump it on a log or get poked by a brier, but it’s hardly more serious than a human hiking in shorts. The tendons on the back of the leg, well, that’s where you get into the problem of either 1. boots not actually providing support or protection, regardless of what the manufacturer claims and 2. boots actually causing damage to the tendons (by heat, pressure, etc).
Polo players use stuff on the lower leg while playing (7 minutes at a time) because the leg may be struck quite hard by the mallet, ball, or another horse’s foot. Cross country competition and schooling uses boots for a short period of time, on a groomed course, to protect against a very hard whack against a solid jump. Barrel racers use boots in case the horse kicks itself hunkering down to make the turn. Pleasure riders use boots because it makes the rider feel better. 
I just thought your discipline probably runs into the same stuff that Trail riders do. Sticks, brambles, etc. Or maybe Hunters main terrain are fields? Just a guess.