I disagree completely. I see it totally the opposite. Maybe it depends on the specific game, but a lot of fighting games are made to be easily digestible and easily picked up by anyone.

Yeah, you're not gonna get a lot better in that time. But that's kinda a moot point. OP has about a couple hours a month. I'd say you'd be hard pressed to find any game that you can actually genuinely improve in with that kind of time.

They're super straight forward these days. And there's tons of single player options and the difficulty can be scaled entirely to your skill level. I think I'm pretty decent at fighting games now. And it's not because I had the mindset of improving or anything. It's just that the games are so simple, fun, and easy to pick up that I was playing a ton just because it was fun. Idk, it's not for everyone but I'd say as long as you're not playing online, it's worth a shot here.

Your sessions sound so few and far between that it would make it hard to play pretty much anything that's about a story. Or where you would need constant reminders about how the gameplay works.

Honestly it might be hard if you can't find demos or trials, but I would recommend maybe trying to find a casual genre that you enjoy playing. What comes to mind is maybe fighting games like mortal Kombat, racing games, or maybe a sports game.