I think the problem is as you mentioned above, the need for damage. See we've been put in a position where any champ you pick in any role absolutely MUST be able to do substantial damage in the early/mid game or they are not playable. Which is why you have Alistar & Leona able to do 90% of an ADC's hp all on their own pre 6, or a Janna poke you for 25% of your hp with just W and 1 auto during lane phase. It's also why tanks were previous overtuned, because it wasn't enough that they were tanky, they also had to do damage.
So here's the solution. Don't touch damage at all for the damage heroes, let them continue to be the damage sources. But then reduce the need to stack damage and increase the value of utility. One of the biggest problems with tanks right now is that they don't really feel like meat shields until like 2.5 - 3 items, which is also right around the time crit ADC's come online, who kill tanks.
So tanks should be allowed to become tanks earlier on. Like a full tank Maokai should feel comfortable initiating a 5v5 front to back team fight at like the 10 minute mark, and you balance this by making it so he doesn't do hardly any damage. What's the point then? The utility is the point. So the buffs that should come into place for tank should not be a damage rune, but a defensive rune like removing aftershock but then adding a flat armor/mr rune to the row with demolition. Reliable tank stats from level 1, and then you buff the utility. Make Maokai's ult root last for 1 second longer. Make brittle targets from Orn's kit do 10% less damage while they are brittle, targets stunned by Ali's E should grant hp regen when an ally hits them. Things like that. Tanks should not be damage sources, they need to be fully viable as meat shields and controllers of damage. If the carries are the water, the tanks are the lock and dam system.
How is this balanced? Well pretty much if you choose tanks, then you are limited to having to choose high DPS carries over things like assassins in most cases. Think about a 5v5, if a Maokai root lasts 1 second longer on all 5 champs, it's not going to do a whole lot for his mid Zed cuz either way he's pretty much going to run in and nuke 1 guy then be done (and remember in this scenario Mao is doing garbo damage himself). Or like Akali will run out of ult charges. So this actually promotes balance as you have to choose between damage heavy comps which can burst guys out, or getting a hefty vanguard for your sustained damage back line. It encourages both styles of play, accentuating the strengths and weaknesses of both.
It might also help curb what I think are two of the biggest problems with the game: an overabundance of mobility, and the lack of ability for ranged DPS to be able to dish out damage as they kite backwards.
