I find it really funny how

Some late game champions are relevant because of the expected payoffs on Kai'sa.

Vayne has a pick rate of 14%, which at first glance means that she's a meta pick. However, this is in large part BECAUSE of Kai'sa's existence.

Let's look at Kai'sa vs Vayne.

2 reasons Vayne has a higher winrate than Kai'sa:

Kai'sa can hold her own early game, but Mid-Game she isn't as good as Vayne, due to Vayne's fast Qs and W proc damage, and Vayne's R's influence over skirmishes. Kai'sa's reliance on snowballing and then hyper carrying is unmatched if successful, and can single handedly decide games by 15 minutes or less.

Kai'sa is played nearly 40% of all games as the ADC. It's closer to 37%, but if you go onto OP.GG, you can see that her pick rate is 37%.

So if Kai'sa is so good, why is Vayne even played at all?

First of all, team composition actually matters a lot, even in low elo. A Malphite Ult + Yasuo Ult can single-handedly win a team a game in just one good teamfight. Take that power, and use it in pro games, and team comp matters even more.

Kai'sa and Vayne have many similarities, in that they're both very very good max hp + burst damagers late game, with Kai'sa being better against enemies with higher health and Vayne better against enemies with higher armor.

Having said that, Kai'sa is usually either banned or picked. If a team is going late-game oriented, then they may want Kai'sa.

Alas, Kai'sa was banned. What now? The most similar ADC in terms of power spikes is Vayne, trading early power for middle power.

Because of this middle power over early power, some people may even prefer Vayne over Kai'sa.

There is one more reason Vayne is played so much more than other ADCs.

If both teams want to play Kai'sa, then the one with first pick will usually lock in Kai'sa before the other team can. Unless, of course, that team picks Lucian, another HUGE player in this meta down bot.

This is another reason Vayne comes in. Vayne stands up to Lucian better than Kai'sa can, and still has that late-game hyperscaling potential that Lucian doesn't.

One of Vayne's applications is a strict Kai'sa "counterpick", but really it's a safety net where the only ADC that can match Kai'sa's late game is picked. No one else can match Kai'sa's sheer late game power. Of course, if a team is expecting a Kai'sa pick, they're going to ban Lucian and other counter champs, making a Vayne pick necessary.

TL;DR: Vayne is good right now because she does what Kai'sa does with slight differences. However, most people prefer the high-octane early game playstyle of Kai'sa to the safer early game of Vayne.

I would expect Vayne to be played less than Caitlyn if Vayne and Kai'sa weren't so strikingly similar.