Everything can "break the game" when you give the players enough freedom. So either you restrict the freedom of your players heavily, turning 5E even more into a board game than an RPG, or accept that there will be no perfect balance.
Even more, you can only "break" the game when you have a very specific vision of how the game should be played (most often tactical combat dungeon crawling) and allow no deviation from it (either because of preference or lack of rules).
As soon as you expand the scope of the game it is much more difficult to "break" as it consists of more than just one activity.
I'm not sure if this is a dig at my assumed playstyle or not, but a player ability can be just as problematic if it has nothing to do with combat. If a feat is available called "Most Interesting Adventurer In The World" and selecting that feat means that NPCs automatically fawn over you and beg your friendship, it's just as broken in a game design sense.
Toward the end of my 3rd edition days my group had several instances where a character's player felt guilty hogging the limelight with everything their characters were capable of, and retired the characters or worked with me to houserule the class into a more restricted form. In 4e there is still optimization of course, but so far it doesn't allow for the excess we saw back in the previous edition. I'd like to hope we can have the same experience with 5th edition.
This branch of the conversation is drifting offtopic so it probably better belongs in the "Every option must be worth taking" thread.
I'd also like to note that nowhere in my posts did I deny that monsters with player class levels should be available as option. I'm perfectly fine with that, but I believe such onerous NPC advancement should not be the default method.