Why not go tank in every lane?

Except they don't have the damage, lol.

Why not have an adc/support bot lane, and an apc/assassin mid lane every game?

Oh wait...

Tanks lack damage vs other tanks. Tanks lack pushing power, and some champions make tanks' lives so horrible that if you pick a team with only tanks vs them, you will have a very bad time.

Because A: They lack the damage to take down tanks, turrets, or epic monsters under a reasonable amount of time, and B: they take some time and money before they're indestructible. They're still fairly vulnerable during early game.

As long as you still have an ADC it's sort of doable, depending on what you consider a "tank".

Cho mid, Sejuani jungle, Renekton top, and Alistar supporting for a Vayne would probably be a pretty solid team IMO.

Because you have no damage. You will get destroyed by any team that knows how to protect their ADC:

As said: Balance, in all things.