Depends how you define ability resets.
If you mean the full cooldown refresh that Kat/Kha/Trist/etc get, then those are kept as limited as possible because of the exploitation factor. On assassins, that refresh is often necessary because their engages tend to also be their escapes. It rewards the assassin play pattern: do it right, get the kill, get away. Didn't get the kill? You are now out of position, and are probably going to die.
The reason full refreshes aren't as common on non-assassins is that if you have this happen in teamfights, and a strong ability becomes available again each time the champion gets a kill or assist, suddenly this champion turns every slight advantage into a win. Again, look at Katarina. Imagine if Vi or Xin Zhao had that.
There are also "lesser" ability resets, where the cooldown is reduced by a certain amount for performing an action, or for using the ability in question a certain way. We see this a lot with champions who want to deal consistently damage over a duration
or who are encouraged to use certain abilities in certain ways
. This means you can have a character who is very, very powerful if you can reliably accomplish their goal (constantly using Lucian's passive gives you a ton of tactical mobility, picking up lots of Q kills as Annie in lane gives you more frequent stuns) but it still leaves room for failure.
TL;DR: An ability with a 20-second cooldown is usually very, very strong. An ability with a 2-second cooldown is usually very weak but spammable. An ability with a 20-second cooldown and a reset is the kind you should be given as a reward for playing well, and not all champions need (or have room for) that sort of explicit reward mechanic.