Riot has a way to balance all champions at once, but they aren't using it
ITEMS!
items are bought by every champion in the game, and most champions have only one build per role. The exception are bruisers, who can build whatever the heck they want and get away with it
if you want to balance champions, increase item variability, instead of every APC building 
, 
, and 
Yes, DoT mages can build  and Mage/Marksmen build , and things like that, but most of the items listed above are core for most mages, every time
Same with ADCs, the only choice i really have is "Do I buy 
or 
" which is really just whether you have mobility or not, and "Do I buy 
or 
" which is really just whether you have AD scaling skills/inbuilt AS.
I kept my ace of spades for last: 
. The jungle would like its champion pool back. EVERYONE runs this, even to the point that people take smite top just to get this. which is stupid.
Riot could make changes to item sets, buff certain items, nerf others, and see a change in the champion pool immediately. But you know, its just harder to do than champion balancing. Because when you change a champion, only they are changed, when you change an item, everyone who bought the item, and everyone who now buys it, are changed/. This is a double edged sword, you can make drastic item changes and is could cause an increase in playable champions, or you make unintended buffs to already powerful champions, and decrease champion play ability. But I feel like that is a risk we need to be willing to take.