Big reason why tanks are nearly gone is because other classes have been powercreeped hard
Especially ADCs and Supports, the new ADCs and Supports have some of the most overloaded kits ever released (Thresh, Kalista, Braum, etc)
Back in 2010-2012 ADCs weren't nearly as mobile or capable of just beating tanks 1v1, and back then supports weren't capable of peeling entire teams with their skillsets (except for Janna). Since then we've seen abominations like Lucian/Thresh/Kalista/Braum who have made it so 1) You don't need a main tank in order to help protect your carries because the newer supports are so overloaded with peel, and 2) even the newer carries can just straight up escape or duel champions if they have a red buff (and they always will).
ADCs slowly have been powercreeped harder than any other class in this game. Whereas other classes like fighters and tanks have seen power decreases as a whole over time (Malphite used to have 60% AS slow and a 70% armor ratio on E, Old Olaf, old Shen, Mundo used to have 15 seconds lower on ult CD and 10% more health healing, old Irelia, etc), ADCs have seen INSANE levels of power creep over the years. Ashe used to be considered the BEST ranged AD in the game by a wide margin, and she's received NOTHING but MASSIVE buffs to her since Season 1, yet now, she's considered subpar. Why? Because every time a new ranged AD comes the entire class gets powercreeped to keep up with them. See the abominations that were Release Vayne, Release Graves, etc. It's not just Ashe either, go look at Tristana's patch history before Season 4 worlds and you'll laugh at how hard she's been buffed over the years. Riot NEVER buffs fighters or tanks except on very rare ocassions, yet they'll buff an ADCs mana costs or CDs or base damage without even hesitating. Tanks can't keep up.
A big part of this is because since the end of Season 2 assassins have been wrecking solo queue and competitive games on and off so Riot had to create champions to be able to deal with the mobility of assassins, but that's beside the point. Basically, the main reasons to pick a tank in old LoL were:
- Serve as a protector for your squishies
- Initiate fights
- Tank damage
All 3 of these have pretty much been nerfed to the point where tanks aren't dominant anymore. There's no reason to pick a tank to protect your carry because the current carries simply don't need it. They either get all the protection from their support being so overloaded or from their mobility, or hell even from red buff.
There's almost no reason to pick a tank to initiate fights for a few reasons. First one is that assassin pick comps usually make it so 5v5 scenarios aren't favored (which tanks thrive in), and the second one is that other champions can initiate fights just as well. Orianna + "insert melee champion here," Syndra Stun, Kalista Ult, Sivir Ult, Jarvan, well you get the point.
As for the tanking damage point being moot lately, well, I don't need to convince you guys of how hard resists have been nerfed and how strong LW/VS/pen in general has been for years now.
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Basically, while AD carries take out tanks at relatively the same pace now and then, additional damage from AD bursters (who were gifted with the ability to ignore over 50% of their target's armor between Last Whisper and Black Cleaver) and from mages (against whom the defenses took a heavy hit) heavily contributes to the tanks perceived short duration in fights.
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