"Training with R+" relies upon controlling access to reinforcement. Controlling access to reinforcement is a learned skill -- it is something human beings take a considerable amount of time, education, and effort to learn to do. Because it's something you have to learn HOW to do, before even beginning to learn how to do it WELL, it can be really hard to understand how it works in practice. Maybe especially if some of the videos you're watching show (a) other people who are still learning (meaning they may say they are doing one thing, but the video may not be a great illustration of their claims), or (b) only part of the learning process. But your question is a common one, Jagger, and your confusion understandable...science-based dog training takes a long time to figure out, I think, because it requires people to do a lot of learning.
Kmes makes a good point that not everything that involved treats is "R+ training." If people are trying to help a dog overcome fears, they are usually employing some degree of counter-conditioning, which is an associative learning process like kmes described.
Fear-based behavior is very challenging. Unless you have lived with a severely fearful dog, I doubt you have any idea how challenging (and by "you," I mean everyone on this thread, not any one person in particular!). Fear is one of the most powerful emotions out there, and the fact that some individuals, for a whole variety of reasons, are more prone to extreme fearful responses, creates behavior challenges that go far, far beyond a simplistic notion of "cures." If any professional trainer claims to be able to "cure any dog," I know immediately that either they are lying outright, or their understanding of dog (and human) behavior is frighteningly lacking.
With that disclaimer in place, here's a straightforward example of "curing" a dog via counter-conditioning. This is an example of changing emotional reactions. A lot of "problem behaviors" are driven by the underlying emotions, and if we can change those emotional responses, we set ourselves (and our dogs) up for success when it comes to changing behaviors.
Background: I adopted my dog when she was about nine months old, and at that time, she found garbage trucks scary (i.e. the sound of a garbage truck made her bark, tuck her tail, shake, and exhibit similar signs of fear, and it rapidly worsened over a short period of time). I made a training plan for her: I planned to use counter-conditioning to change her emotional associations with garbage trucks.
First, I found a treat that my dog felt REALLY good about (cheese). Then, I made sure that the sound of a garbage truck was immediately followed by a piece of cheese, every time it happened. It helped that garbage trucks in my neighborhood went past at pretty predictable times, so I could always have cheese ready, and that they went past frequently (four days a week, so there were a lot of chances to practice!). Not a controlled environment, but one where I could maximize my ability to use "random" events to my benefit.
This was purely associative learning: my dog learned that (event 1) the sound of a garbage truck always predicted (event 2) the arrival of cheese in her mouth. In short order, my dog's behavior changed: instead of reacting to trucks with fearful behavior, she reacted by running to me for cheese -- dogs are really good at learning predictive relationships, and my dog was learning quickly that trucks predicted cheese. The stronger this predictive relationship became, the more confident (less fearful, less anxious) her behavior became. In other words, I could see her emotional response to trucks change from "scary, bark bark bark!" to "oh boy, CHEEEEESE!!!" A few weeks after her emotional response to trucks had changed to happy, confident expectation, I began gradually cutting back on the cheese (watching carefully for any signs that the fear was coming back, and making adjustments as necessary).
Today, I live in a different house, and large trucks actually pass by pretty frequently (not on a schedule, and I can't predict when...there are garbage trucks, buses, delivery trucks, and more). My dog doesn't even notice them -- she doesn't wake up, flick an ear, or run to the window barking, let alone show the kind of fear she used to. That's as close to "cured" as I can imagine. And no, to answer your question, I do not need to perpetually stuff her with treats to maintain this result, for a whole host of reasons that would make this post unbearably long to type up, although I did use a second, brief period of counter-conditioning to help smooth the transition when we moved houses.
To claim that all fears can be dealt with in exactly that way would be extremely simplistic of me (not to mention insulting to the thousands of caring dog owners who've tried). But in this instance, a few months of counter-conditioning and several ounces of cheese were more than enough to drastically re-wire my dog's emotional response to something she used to find pretty upsetting. And it was extremely low-stress, humane, simple (which is not always the same thing as easy), cheap, and straightforward, so that's basically a win-win everyone involved.
If you're talking about simple obedience behaviors -- things like sitting on cue, other tricks, or a stop-on-a-dime recall -- then the "paycheck for life" question has a more complex, technical answer. In a nutshell, no, not every behavior gets reinforced constantly for a lifetime (though many might, depending on how strong you want the behavior to be, and what forms of reinforcement are available to you), but training to that level takes a considerable amount of practice. And more to the point, why wouldn't you want to reinforce your dog for excellent performance? My dog does brilliant things that make me enormously happy, so why wouldn't I want to "pay" her for those things, whether it's with some portion of her daily nutrition, some portion of her daily affection/attention/engagement/etc., or whatever else I have handy? Not to mention, there's no point at which my dog is magically going to stop learning, so I might as well keep making sure she learns the lessons I want!