Remove CleanMyMac immediately as it usually causes more problems than it solves. The League of Legends support team should be telling you exactly which items they want you to remove instead of advising you to use such an untrustworthy app like CMM. Use the CMM developer's instructions on how to properly uninstall CMM.
If you want you can run EtreCheck and post the report here using the "Additional Text" icon which looks like a piece of paper. This will let us see if you may have any other software which might be interfering.
Have you tried just rebooting your computer?
One safe way to delete some temp & caches is by booting in to Safe Mode.
Have you checked to see if Gatekeeper needs your approval to allow League of Legends to run? Check in the Security pane of System Preferences.
Create a new user account and see if you can launch the game while logged into the new user.
Have you tried completely uninstalling the game according to the developer's instructions, then rebooting, and downloading and installing a fresh copy of the game?