As someone that fits some of the above categories, I think you really have to step back and repeatedly tell yourself "get over it". Its the same mentality to "I dont want to go to the gym today". You immediately feel better as soon as youve finished it and wonder why you always drag your feet before.

This becomes an unlimited excuse.

Even if there were state programs that established and ran these sorts of events and created low-friction ways of interacting with people, people could still say "well that assumes a certain baseline of energy."

It is true that somebody who is in the midst of extreme depression and can't get out of bed is probably not going to be able to set up a local dnd game. It is also the case that the large majority of people are absolutely capable of doing this sort of thing.

If I may suggest to start small, it doesn't have to be a group of people playing football. I personally like to just meet 1 or 2 people to which I can have interaction with all of them.

If I may I made an attempt to crack at this very problem with Tatapp (tatapp.astekita.com). Any feedback is very much appreciated.

I agree. I also don't think forcing yourself to be an organizer is necessarily a solution to fixing the loneliness, as it also just requires a certain passion. In my experience, some people love organizing things, others just really hate it. I am in that last camp, after having organized quite a lot. For me, simply participating with things that are organized by others has done me much more good. Of course, that still requires being in a state of mind where you are able to take initiative with signing up for such group activities.