TL;DR Help a guy find a champ pool to turn around more games when they don't go so well
This is less about a champion pool and more about a player. I guess the best champions for this is teamfighters that don't require a lot of gold. Teamfights win games far more often than they should.
There's another component and that's being able to take out the champs that are carrying the other team or is at least fed.
However to turn a game around, your team needs to be able to fight as a team and win, even with a gold deficit. That's not a champ, that's a skill. Even if you go 1 for 1 with the player carrying their team, if your team can't teamfight and theirs can, you guys are still screwed.
So maybe you'll think "I'll just splitpush". It isn't that easy. Especially when your team wants to initiate a 4v5 without you. What usually happens is your team will start a 4v5, die without getting a kill, and you are screwed. Worse yet, they won't know how to delay, disengage, and keep the other team's attention while you split push so even if you are lucky enough that they don't start a lost fight, they won't be able to prevent one either.
My best advice is learn how to win your lane and then learn how to team fight. Learn how to start a team fight and learn how to keep one from starting.
A lot of the champs I play I manage to win my lane. My issue is even if I go 8/0, most the champs I play cant carry hard enough to turn around games where the rest of my team didn't do so well.
Example - One time I went like 10/0 against a Riven as Chogath, absolutely demolished her... she couldn't even farm under tower without me killing her, and the jungler also if he bothered to try and save her. At one point she rage quit, and the game became 4 vs 5. Even though I was extremely fed, had high cs and items... I couldn't carry everyone else that did very poorly.
My cooldowns were too long, their team work was really well organized and they focused/tried to catch me out a lot, and my team was like a bunch of paper bags to their damage. Now I have watched games where Akali is behind, or leblanc, or katarina... and a double becomes a triple... and then a quad. Next thing you know she is penta killing the enemy team on her own.
Again TL;DR - I know skill is involved, but there are champs that carry harder than others... that's a fact.