That still doesn't solve the problem I proposed however.
I proposed that assuming CDR is no longer a stat on items and only through abilities will be granted that if you ever tried to do ANYTHING that wasn't building specific items in a specific way
that it would result in you losing the game.
You said it yourself that changing the item affects every other champion that builds it...
So if you change the core item of almost every mage in the game
you change a giant portion of the game to not exist.
You also change the core item of almost every bruiser 
and change the core item of a few adc's 
Removing variables makes balance EASIER... at what cost however?
Keystones have made the game more interesting, Sure there are times where riot failed at balancing keystones not to mention riot also recognized a good portion of problems and fixed them for the time being(respectively at the times of keystones being overpowered)
I don't want to play a game where I have to endlessly poke someone for years in the midlane with
VS 
I don't want to take forever to fight someone in the toplane in a
VS 
I just realized something...
You would make marksman insanely prominent in the game.
Basic attacking has no cooldown...
as a result there will be little opportunity for mages to exist because marksman do more DPS on NO COOLDOWN
Tanks will suffer as a result and won't ever be played thanks to marksman that counter tanks
and 
There wouldn't even be space for a jungler because having twice the marksman means having TWICE the anti-everything. Not to mention if one marksman+sup fails in the botlane then the toplane marksman+sup will get attention from the midlane marksman.
Do you see the problem here?
I do.
You're next best argument would be to say "well then if we make those changes we'll just reduce the attack speed of marksman items"
and you're right.
You know what would happen after that?
People would stop playing marksman and marksman would be replaced by tanks
Marksman counter tanks due to their nature of kiting and large amounts of continuous damage.
If you take away the one major role that can deal with tanks...
You've created a tank meta. Congratulations.
I don't know if you realize the consequences of nerfing things
If we buffed weaker champs, items, etc. we could easily come up with a "everyone is viable" type of meta.
If you need further explanation of why taking CDR away is bad then reply to this post with "yes"
if you want to debate further on the OP then reply with "no" and your argument