I have to put my moderator hat on here briefly.
I don't know who the rider is in the second video, but if she did not consent to have the video posted here and critiqued or used as an example, I think we should refrain from commenting further.
Moderator hat off, and back to our conversation.
I am a big fan of William Fox Pitt, and I will see if I can find something of his to post. I love moments such as you have described, because they are the proof of training technique - if you have always ridden your horse held precisiely between your leg, seat and hand and never independently, freely forward you get in those sticky situations and your horse does not know how to find a fifth leg and save you. You can get away with that in the jumpers, though it will never be a style I admire. However, in both foxhunting and XC, your horse must be secure and confident enough to continue to go forward and where he's pointed in those "Oh, crap!" slip the reins sort of moments.
PS - the horse in my avatar had a much more successful career as a jumper than an event horse as dressage was his forte. We frequently disagreed about pace and balance going into combinations; which mean we frequently jumped in really, really wrong, at which time I would slip or drop the reins and say "Okay, smarty pants, you got us in this mess, you get us out." and we not only both lived through the experience, we won our share. A lesser or differently trained horse would stop, pull a rail or crash. (Okay, and a better horse would have listened to me, since I walked the course and he didn't, but we eventually sorted that out.)
I think that any horse, especially as they go up the grades eventing, thinks that they know better at the start of the course.
Watch many of the video's, including the one posted here earlier and that of William, and they are both, as are many others, having to demand rather than ask, over the first fence.
I agree with you about a bold horse. Very important around here over the hedges and trappy fences we come across.
The advantage of hunting here is that it does give a horse a good grounding for jumping off all sorts of terrain.
One of our show jumpers took his top international show jumper out hunting several times because the horse was unsure jumping if the ground was wet/muddy. The horse loved it and the problem was solved.
One of the best rider CC was Licinda Green (nee Prior Palmer) I watched her jumping the Trout hatchery at Burley. The horse made a mistake and nearly went down. She just sat there giving him his head and somehow the horse recovered, jumped the fence immediately in front of him and the one out of the water.
Lucinda rarely rode a horse in a martingale because if things like this happened they needed total freedom of their heads to recover.