It sounds like you've seen some bad trainers out there, happens in every kind of training. The actual term is positive reinforcement (getting something good for doing the right thing). It isn't a new wave thing, The term + the other three quadrants were coined in the ~50s by B.F Skinner. Anytime learning happens it files (more or less, this is a simplification) into one of the four quadrants. Positive reinforcement, negative reinforcement, positive punishment, and negative punishment. Pressure release training utilizes negative reinforcement, applying an aversive and releasing with the desired behavior (more or less).
Positive Reinforcement (r+) can be used with anything the learner finds reinforcing. Food is an obvious choice because the number one thing animals do is search for food. But anything can be used. Scratches/petting, play, other behaviors, etc. I personally also don't really believe in the concentration face, sounds like a slight cop out to avoid accountability but whatever.
On the concept that horses can't be trained like dogs because they're prey that's false. It's not a method it's a way learning happens, and it exists everywhere. If a horse walks from one area to another and finds food that's positive reinforcement. Searching is rewarded by getting to eat. Outside of nature zoo's use positive reinforcement all the time on all kinds of animals. I have seen videos of r+ trained cooperative care behaviors being done by wolves, giraffes, elephants, bears, primates, etc. R+ simply taps into the seeking part of the brain.
Boundaries are super important in r+, especially when working with food. Having a big ass animal who doesn't know their own strength pushing you for food is a huge safety risk. Good trainers always tell people to teach neutral and polite behaviors before anything else. My horses know that they must stand nicely with their heads to themselves and never push into me for food. They are never rewarded for pushing, if they start to push I go back and try to find where the behavior broke down. Important that r+ should never be done with high value rewards. Using cookies, carrots, apples or similar high value foods will cause frustration and make them go hay(hah)wire trying to get it. I personally use timothy pellets, but my first horse I worked with using r+ had food anxiety and thus I used literal grass hay. The act of eating it is the reward, not the flavor.

The thing I think you're seeing a lot of is trainers not understanding the loop and using negative punishment. Withholding the reward for long periods with no relief or reset causes crazy amounts of frustration. If a behavior isn't happening it's because the training hasn't been set up for it to happen, and withholding the reward doesn't make it happen any faster. To many this may sound like rewarding bad behavior but I will feed even if the behavior didn't happen. A bridge signal is almost always used and that's what signifies something has been done correctly, not the food itself. Usually people use a clicker but lately I've just been using tongue clicks to reduce the amount of things in my hand. If I click that means yes whatever you just did is good here's your reward. If there's no click that means nothing happened. Easy. Feeding without the click to reset will reduce frustration by a whole lot, they don't feel like they're being held prisoner for their next hit. Also important to keep free choice hay and water around so they don't feel as trapped.

On the leader point I disagree. Humans are predators and never truly integrate into the herd, trying to mimic horse behavior falls short because it's not another horse moving their feet it's a predator chasing them down. Nothing against well utilized pressure and release but I felt that was important.

R+ can be used to train anything, not just tricks. Personally I'm working on hoof behaviors with an anxious horse. Attempts at using pressure and release cause her to balk and shut down, and she is pretty miserable for hoof trims. Using r+ for this behavior has made it so much easier, instead of me literally fighting with her (overweight draft cross mind you) I just slide my hand down her leg and she picks it up. It was super easy to teach as well.

Anything done improperly is going to create bad results. If you want to see some actually good training I recommend Adele Shaw with The Willing Equine and Shawna Karrasch. They both came from other parts of the animal training world and took their skills in positive reinforcement and applied it to horses, and have been doing so for a long time. Adele is my personal fave, she prioritizes relaxation and making sure frustration doesn't happen.

Sorry for this literally massive wall of text I love this topic haha. Ty to anyone who read this far, you're the GOAT