I think it is much more nurture than nature.
I think maybe 75% of the time, a hot forward riding horse is that way because of nurture and not nature.
I have seen so many horses come to my trainer’s program as “go” types. They have varied from well trained but very forward eventers and jumpers to mishandled OTTBs off the track who are running and nervous under saddle.
Out of probably the 175 horses I have seen go through her facility, I can think of maybe 10 that stayed as forward, go, hotter rides after being given tons of turnout and low key jobs. I think most horses, if they lived outside 24/7 and were given a low key job, would go from hot rides to push rides.
I am NOT saying that anything hot/forward his been mistreated/not given enough turnout/etc. When I was managing a show jumping barn, a lot of those horses were hotter and forward. They went to a show or a jump school and were on their toes, ready and eager to perform. But at home, hacking around the ring or a field, they were pretty much all what I would describe as “push” rides. All the well trained “hot” horses I have ridden have ridden that way because they have been trainer to be forward, they have eager personalities, and they genuinely enjoy their jobs. Not a bad thing at all!
I’ve ridden event horses who I needed big spurs on to get them active and engaged on the flat at home and then were certainly forward cross country. I’ve ridden fox hunters who do walk/trot lessons at home and then hunt first flight in a gag bit.
I guess my barn’s unique situation made me think a lot of it is nurture and environment instead of nature. I just saw so many horses come in jacked up, scary hot under saddle (as well as hot, well trained performance horses who needed a step down job), who after getting turned out 24/7 and doing easy lessons a few times a week, settled in to a new routine and became some very quiet, lazy horses.
I am NOT saying that 24/7 turn out and a low key job is the best fit for every horse, just that I have seen it drastically alter how a horse rides under saddle. In fact, one of the horses I think of that is still hot and forward is a mare that was donated after a long career as an UL eventer. Every ride on her is like she is back cross country - she is forward, bold yet responsive. She probably would be much happier still being an eventer, but she wouldn’t have stayed sound.
So personally, I think nurture/environment/rider expectations have the biggest component to it and that the number of horses that are truly “hot” is actually a lot smaller than one would think. There are horse that have it in their nature, because I’ve seen horses raised on the farm the exact same way and home out hot and forward, but there have been so few that I think nurture is such a big part of it.
But perhaps nature breeds a personality and athleticism that lends itself to upper level training, where a horse is going to be encourage to be more forward (thinking specifically of show jumping and eventing here.)