There were lots of times Anivia threw a blank wall and put it on cooldown.
Or was at base, replenishing mana.
You don't need minions to push a turret.
Lee Sin could have tanked 2 turretshots, and shielded back to his team in case of danger. Corki could have walked in, hit once, and Valkyrie right back up and wait for it to come back again.
They didn't even need someone to tank it. 2 Corki autoattacks with Trinity proc would kill that turret. Easily. They were just too scared. They had lots of opportunities to harass that turret, but they didn't take them.
Hell they could have basically made Nunu march down there and went ape**** on that turret, and still win the game due to Gold + item + Baron advantage.
If they did one of those, we would just say, "well, Anivia did all he could but couldn't hold them off for so long" and we would be right in saying that. Froggen really did give them some windows of opportunity at certain points, but none were taken by m5.