I agree this is very different from normal UK BHS teaching. But feeling the horse is what I was taught to do by the trainer Mark Rashid and it is the root of my riding.
The question is what do you feel? And has anyone suggested what you should feel for?
Rashid usually starts by explaiing the beat of each gait. So walk is a 4 times beat and as the horse walks you should feel each of our ips drop (alternately) as the hindlegs of the horse move.
Lots of people think that when your hip rises, that is the point the horse's hind leg rises too. But in fact it is the opposite. Imagine the horse like a table with a leg at each corner. If his right hind leg is off the ground, there is nothing supporting the table at that corner ad it will dip.
When the hind leg of the horse is off the ground, that is the moment you can guide it. For instance in leg yield. But knowing which leg is off the ground will also help you to time your cue for canter.
The movement you feel iunder your seat bones in trot is the same as in walk, but it is now 2 beat.
Feeling the horse move under you and understanding what you feel means that by relaxing or stiffening your seat you can slow down or energise the horse. Rashid taught his studients in a circle in a school to transition up and down simply by thinking the new rhythm,
If you relax and breathe deep and start to experiment with this you will find the horse under you responding and reacting. You are allowing the horse to move forward but you have the choice also to stiffen and thus slow the horse.
It is an additional and reciprocal means of communicating with the horse through the feel in your seat and your thighs. I found this easy to do from the start of my lessons when I could hardly ride at all. My RI told me that some people never "get it". You may be a very good rider but never use this feel at all.
But if you can work it out with the help of this teacher, it is a great thing to have in ones riding repertoire. Going into canter can be magic.