Its fun to play around with in an area, as a demonstration of how a horse works off leg and seat. End of story.
Its not the mark of a good trainer, or evidence of a great horse. Nor is it in any way evidence of a "bond"
I don't know if this is becoming so popular from the notion many people still cling to about the "magical bond" that has been popularized by the king of the wind, black stallion, etc, or what. Its annoying, misleading to the uneducated and fool hardy.
Horses have bad days. Things scare them. Some how being on a horse bare back and bridle less out on a trail ride when a bear decides to say hello doesn't sound great. Nor would a situation I was in last year, when a familiar trail was unexpectedly colonized by ground dwelling wasps. We rode over them, and my mare ended up with 20-30 stings on her belly. I rode it out, but my horse that never bucks or bolts, bucked and bolted.
We had a similar situation a few years ago. In a pasture that I've been riding for 20 years, there was suddenly a bumblebee nest in an area of high grass. We were doing close cattle work, trying to get some cattle sorted off from others, when my Dad rode over the top of the nest and out came the bees. They started dive-bombing us and we all (horses included) got stung at least once each. My step-mom had a bee fly down in her horse's ear. Thankfully, because of his inherent nature, he didn't have a complete meltdown but he darn well could have because that's not exactly something that you can prepare for. Dad and I were both on horses that had panic attacks and were pitching and trying to bolt.
As for liberty, it has it's place to showcase a well trained horse in a controlled area. Even on my best trained horse, I wouldn't dare take him out without something on his head. It's not that I don't trust him, but I know that horses are naturally flighty critters and if something were to happen, I like knowing that I can get control of him. Have there been entire rides where I left the reins draped over the saddle horn or laying on his neck? Yeppers, but they were still there had I needed them.
About as close as I ever come to riding at liberty is bringing up a horse from the pasture bareback with nothing but a string around his neck. If he chose to (or felt he needed to) take off, I couldn't stop him, but I'm in a relatively controlled environment and they have training that's good enough that they'll listen to my seat and that string in all but the most extreme circumstances.
Just because a person can
do something doesn't mean it's something wise to do. The example the OP posted shows exactly that. The rider was more concerned with "ZOMG, look at how cooool I am that I can ride without tack" than the fact that she was responsible for taking care of the novice riders following her.
I've had to play impromptu pick-up man in the middle of a pasture for folks with bolting or bucking horses. Thankfully, they all had actual tack where all I really had to do was get close enough to dally up a rein, but I wouldn't have even attempted it if my horses hadn't been in full tack.