I completely understand that! It throws you into a mission right from the start, with no easing in. And from the very beginning you can see that the game allows you to choose - do you kill? Do you sneak around? Do you shoot the leader? Do you let him go? What are the consequences of your actions? It's not all clear from the start.
I played the original Deus Ex around 10 years ago. Once you accept the aged visual side of it, the world, characters and plot can easily immerse you till the end. I'd say play the original (maybe with some modern visual mods), then skip to Human Revolution, then Mankind Divided.
I didn't bother with Invisible War and The Fall. I tried them for a moment, but the bugs and clunkiness put me off.
There's also Deus Ex Go, a mobile game that's actually a really good puzzle with fantastic music.
If you especially liked K-On!, then I guess Lucky Star is the closest one, with no music band stuff. Note that LS has tons of references to other media, so if you're new to anime, 80% of the jokes will fly over your head.
Nichijou is very comedic, and so is the other one with Nichijou in the name (they are unrelated, just the generic term "nichijou" means "daily life").
The Aria series is my favorite; it's very slow, to the point of being boring, but you grow to really love the characters and end up bawling your eyes out when it ends.