I came across this quote yesterday:
“If you go out looking for friends, you're going to find they are very scarce. If you go out to be a friend, you'll find them everywhere.” (Zig Ziglar)
It was in a book about influencing other people, rather than making friends, but it's one of those things that rings true for me, and matches up with how I've felt in social situations.
For me, "putting myself out there to make/get friends" is weird, cringeworthy, and scary. Like going to one of those dating events where the only point of it is to hook up with somebody. And their reaction to you is either "yes, I'm instantly attracted to you," or "no, go away, next please". It doesn't allow for the common situation where friendship develops over time, and it's going to be pretty depressing for the majority of the population who don't instantly "click" with someone because, statistically speaking, you're going to get more "no's" than "yes's".
I have never found friends "everywhere", but I've always had more success (such as I have had, and my Christmas card list is not a long one) with putting myself out there to do something other than make friends. For me, friendships are rare, but I have friendly acquaintances.
Joining a group (club, organisation, whatever) where the point is to do an activity puts you in the way of people you already have a common interest with, and whom you can get to know slowly. And there isn't the sense of instant rejection if you don't make a friend/click with someone first time out because, hey, you're there to collect stamps/play chess/go skydiving or whatever, not to make friends, right? Actively helping with running the group (even just being the person who's willing to help with the clearing up afterwards, or the setting up, or whatever) is good too - it gives you "social points" and puts you further into contact with people.
And in any group that does something, people willing to help are always greatly valued. In my experience, in voluntary organisations like hobby clubs, 90% of the legwork of keeping things going is done by 10% of the people. So those people are precious!