Give a two opposing teams 20,000 gold to distribute and the one with a tank will have the advantage almost always.
The issue with why the question of a tank being vital is unanswerable in terms of sound logic is that if the team full of squishy assassins has 20k and the team with the tank only has 10k or 15k, the squishies will probably dominate. Even 1v1, my money's on the assassin as tanks typically don't deal that much damage. Assassins can kite, poke, and swoop in for the kill. Tanks are monsters in skirmishes and teamfights with allies who can dish out the damage. So the team full of squishies can easily overpower the other and/or get fed faster and simply outpace the tank team. Then it doesn't matter.
I've been on teams without tanks against teams full of them (I'm talking 3 tanks with a fighter) and we dominated simply because of our insane damage. Twitch got fed, Fiora got fed, and Riven got fed. We also had a Nami, I think. Don't remember who the mid-laner was. I think it was someone with a stun, but it was an AP caster. We cut through 'em like butter.
So while I believe tanks provide an edge -all things being equal, I don't think they're essential. They're just safe. I always feel safer if there's one on my team, but I've won too many games without them to dodge a ranked game because we don't have one.