The Inevitable "Hey! You're using 4+ gems of that!"

Cooldowns are on individual skills, not on your player, so this is currently the correct behaviour. I will being it to the attention of Carl when I see him, but I suspect we might want to keep things this way - while currently all flickerstrikes have the same cooldown time and count, if we ever want, say a "decreased cooldown" support, or anything of the sort, or even a skill that changes cooldown time as the gem levels, that means different versions of the same skill could have different cooldowns, so it makes no sense applying one to the other.
But there may be other ways to attack this issue if it proves a big balance problem.

At least people doing this can't have all the gems supported as effectively as if just using one.

I can't say if it'll change, and that's not my call, all I can say for now is that's how it's coded to work, and give some suggestions as to why it's good for it to be that way.

Consider trying to apply a global cooldown to coldsnap if you have two coldsnap gems, and one is supported by trap - you have two coldsnap skills, and they have completely different cooldowns. To me the intuitive version is the current one - that each skill has a cooldown. If you choose to have multiple of the same skill, each has it's own cooldown.

But it certainly may need changed for balance - so it may change to something else in future.