Being strong early game doesn't mean you outdamage bad early game champions.
This is a common misconception on Boards. Power or (time in game)-strength is correlated, but not proportional, to raw dps or ease of function.
It's related to priority. Priority is the concept of, "What should we focus on fighting with/against first? Second? Etc."
In the early game, it's all about lane/wave priority. Take Viktor vs. Orianna. Viktor is objectively a stronger early game champion, but at level 3-5 if he gets in a raw DPS competition with Ori he will always lose. So what makes him stronger? It's that he pokes harder and has deadlier followup so you need to be extremely mindful of jungle ganks and the 2v2 fights.
What does Ori do if her jungler ganks? She shields him and then QW's. Then autos if she's close enough. She gets a quarter of your hp down and a small slow and then autos. Viktor will slow->stun you and chunk you for 1/3rd to half of your HP. That's much better for a jungler like Lee Sin to beat you down.
And since he's stronger he has priority in how he wants to manipulate the wave. This is the same way Pantheon is strong. Pantheon is strong early because he's like, "Huh. Right now I have priority and I want to keep this wave as close to the middle as possible." so that it's impossible to trade with him if you jump on his minions and he can freely poke you and force you off cs/exp. Same with Kog. Kog'Maw has almost zero lane priority so it's easy to force him off the wave. It doesn't mean if you get in a fight with him as Lucian you're gonna win, which is what a lot of people here think. It just means you're scarier for him to trade with if you play the matchup correctly.
Mid-game strength is almost always related to push/roam priority. If I'm playing vs. Syndra as Ahri I'm now weaker in lane because I can't push as hard but I can roam better, so I'm usually more of an asset at this point in the game. The exception is if I'm forced off the wave because their jungler ganks and puts pressure on me. But in a vacuum, I'm stronger because I can get to lanes more quickly and provide more deadly followup.
Late-game priority is either splitpush or teamfight oriented. TF is strong late-game because his splitpush is godly and in teamfights he can completely change the way frontliners charge the backline so they don't get kited and he shifts the way the support plays to peel the ADC and how the ADC buys items. Syndra isn't as strong, because she can't access the backline without flanking on an immobile carry and you can buy MR items if she's always going kamikaze so she trades poorly. LeBlanc isn't that strong because she can't stop pushes well and it's easy to stop her teamfight/flank. Vlad is strong because he can run into your backline and sometimes kill all 3 members with impunity.
tl;dr early/mid/late-game champs are so because they force you to play more around them at that point in the game than other champions in their class.

<--- still obnoxiously broken.
