Why can't we see our opponents summoner spell cooldown time after they use them?

Gives too much information. It could show to allies that weren't there in the fight and if not there's nothing stopping their teammate from telling them the cooldown left.

Don't get me wrong, the micromanagement of keeping track of cooldowns is annoying and I'd love to not do it, but it's one of the complex things in this game.

That would make the game way too simple.

I don't think that causes people to actually think about cooldowns. Plus, it can be modified by insight or whatever that resolve mastery is. I wasn't a fan of the dragon, baron, and buff timers honestly. Makes the game too simple imo.

A lot of times people actually miss when an enemy uses a summoner ability. Right now if someone isn't sure they have to ask their team mates and hope someone knows for sure if the ability was used or not, which adds a level of skill. Having it on the scoreboard would eliminate this awareness-related skill.

Probably for the same reason we are not allowed to see the enemy's basic, passive, and ultimate ability cooldowns. Riot wants us to outplay our opponents by skill, not just by having all of the information about them at our fingertips.