Very open and honest; that's much appreciated. I sympathize with your way of seeing things, much of it seems rational, and I can understand your desire for a relationship.
Here's what I am now suggesting which might have some impact on you. Being with a girl makes you happy, and it should! I believe this is something that should be desired by any man who is in touch with reality. I just want you to see that the thing you claim to bring you happiness has probably also brought you much sadness over the years. In other words, your lack of having a girlfriend has brought more sadness to your life than the happiness generated by having a girlfriend. So I will meet you halfway with my suggestion. Of course I understand that I am by no means in any situation to dictate any aspect of your life, I'm just suggesting a plan that might work. The idea is to expand upon your hobbies and passions while also looking for someone on those dating websites. In this way, you can get in touch with other things that are 100% sure to make you happy while also trying to find a girl. It's a win-win situation. The point of this being that there are many different passions and hobbies in this world that will make you happy no matter what, so you might as well find out what you're missing out on. You said that you're burned out on playing video games. Maybe it'd be beneficial to take a break from video games and try something more rewarding...
One more piece of advice if you don't mind. No matter how hopeless you might consider your situation to be, just realize that this is your perspective on things. I, for example, see nothing wrong with a guy who has only been in a two month relationship. Every decent human being understands that people go through $hit. That's just the nature of things. It is also Critically important to understand that no matter what your situation is, worrying about it will do you no good. The situation remains the same and you will only be helping yourself if you stop worrying about it. You know that Bobby McFerrin song "Don't Worry, Be Happy"? A line in the song, "in every life we have some trouble, when you worry you make it double", speaks the truth of your situation. Don't bring yourself down with these thoughts.