I'd probably look into fighting games, although of course not all of them would fit in your description.
Something along the lines of Street Fighter Alpha 3 maybe (it's probably one of, if not the most complex and skills-based in the franchise). I hear good things about Guilty Gear XDrd as well.
When it comes to FPS gaming... heh. I wouldn't be very well placed to judge overall since I've never really been that competitive in those, although I did have my "glorious days" during the UT 2003 / 2004 years (that's probably because I was something like 13 years younger at the time (holy sh- ...) and obviously had much better reflexes then. Nowadays it's supposedly "all about" Overwatch, but despite the fact that I've played Overwatch for over 600 hours by now since Open Beta I STILL to this day cannot for the life of me take that game seriously for "competitive" play. How do you compete against heroes that have a wallhack (widowmaker), a legal aimbot (soldier 76), an auto-locking turret that shoots missiles at you at level 3 (torbjorn), a healer that auto-locks on a healing or a damage-boosting beam on their allies (mercy), a healer that's going to suddenly put you to sleep even if you were injected with the power of the Infinity Stones in your bloodstream (ana), a cyborg ninja that keeps zippin' around while cutting half your team in half while not being able to keep eye contact on him due to his insane speed (genji), a hero that lays traps and spams an area for area denial like artillery without any aiming skills requirements (junkrat) ... etc... etc. The answer is: you don't.
I mean, sure it was 'successfully' turned into a new eSports phenomenon (at least mostly in South Korea, big surprise there), but it just wasn't necessary to start with. But Blizzard saw the obvious opportunity to make more money from such events and went for it. But that's just Blizzard being Blizzard, and that's just South Koreans being South Koreans (and by that I mean South Korean gamers would suck Blizzard's dick if it had one and turn anything they'd develop into a new eSports phenomenon). I do really like Overwatch as a generally fun game to play casually to blow off some steam or to burn some time here and there. But there's absolutely no way I'll ever recommend to play Overwatch for competition or to have a fair challenge. And besides, to be honest, Overwatch genuinely has the absolute worst, most toxic community I've ever seen in my entire gamer 'career' online. I've seen bad ones sometimes but Overwatch tops them all and crushes them like insects. On the other hand, if you can tolerate it, sustain it, endure it... then you'll be ready for anything and everything else life can throw at you with a smile on your face. It truly is a test of patience. And, on a side note, don't feel too offended by me judging Overwatch, I genuinely like it, but as a casual online shooter. And the community really is getting on my nerves literally on a daily basis (or each single session I play it), but it's also related to my age, I don't have the patience to deal with immature adolescents and failed adults like I used to 15 years ago.