There are plenty of 'one-trick' players who can make it to extremely high elo even if they aren't particularly competitive. [0]There's a schoolteacher in Korea(?) who streams in a cow costume and plays only Master Yi, on the Korean server. He's Diamond.
And then there's players like BoxBox, who isn't exactly a one-trick, but clearly mains Riven. [1]He's at ladder rank 334 at the moment.
And as a final example, there's the monster himself, Apdo. [2]Apdo is a famous player in China who has on multiple occasions been rank 1 (or boosted people to rank 1...) using mostly Twisted Fate. He's more than talented enough to play any champion well, but is most known for TF. Similar to BoxBox and Riven, but on a whole different plane of skill.
All this is to say that there's no real need to change champions to continue climbing until you're at least above diamond, or if you want to play with a group where a synergistic team composition would be meaningful (like amateur tournaments).
[0]https://www.twitch.tv/cowsep
[1]https://na.op.gg/summoner/userName=boxbox
[1]https://www.twitch.tv/boxbox
[2]https://lol.gamepedia.com/Apdo