Thanks for the detailed description of your experience.
I don't like framing the issue in terms of "missing out". It leads to the "paradox of choice"-type worrying. There are always more interesting and fulfilling things to do than hours in a day. You seem to have developed a good social life framework for yourself that doesn't include Facebook, so I wouldn't say you're missing out.
As for stream, if mine looked like that I'd be unhappy too. You could probably clean up yours quickly though - just scroll through it and/or visit profile pages of some of your friends, and start "liking"/commenting on posts that you find interesting. Facebook filtering algos will quickly catch on your interest, and you'll be seeing much less crap.
RE etiquette, as far as I can tell you're correct; generally you accept people you met IRL, with rare exceptions of both kinds - sometimes you don't really like your new acquaintance, and other times you may decide to add someone you met on-line if you know them well enough already. At least that's how I myself and people I know use Facebook - maybe the culture is slightly different in other parts of the world. But the rule of thumb seems to be: if you think the person is OK and you might want to talk some more in the future, it's OK to send/accept a Facebook friendship invite.
> It just doesn't fulfill any purpose in my life and actively makes me angry.
That's totally fine and, IMO, a perfectly good reason to not use Facebook.