Well the idea behind these champions are to control the early and mid-game for your team instead of power farming. Like, if you want to powerfarm to hard carry, then Graves is probably an option. Elise, Gragas, and Rek'Sai aren't really solo carry champions (they can hard carry games by themselves though), they're more about helping the team. Rek'Sai and Gragas also have access to more creative gank routes.
I mean, I've probably never played any champion more than 400 games. Season 2 I played Irelia, Renekton, Jax, Olaf, Riven, Udyr, and Warwick (I transitioned from Jungle to Top). Season 3 it was Irelia, Renekton, Jax, Olaf, and a bit of Riven? Season 4, it was Jax, Irelia, and Tryndamere for the first half, then Ryze, Maokai, and Alistar for the second half. For Season 2 and 3, I probably didn't play more than 40 games on any champion, few getting close to 30 (not sure if any actually broke 30). Season 4 I played the most I've ever played and had over 40 games on a few champions, mostly 20ish on others. Season 5, I played Maokai, Ryze, and Hecarim early on then went on to Jax, Shen, and Lulu. Season 6, I played Poppy, Trundle, Maokai, Ryze, and some Swain. Atm I've been working on extending my champion pool beyond Trundle and Maokai. I might swap champions a lot, but for about 3-6 months at a time I'm basically a 2-3 trick pony per role I play (which is now 2 instead of 3, because of Dynamic Queue). My point is, I don't really think it takes a lot of time and effort to learn a champion, especially if you're willing to accept a lot of losses early on to learn. I generally jump on the 2-3 champions that are meta (or close to meta) that I'm comfortable with, then grind with them.
If you've played 1k games with Udyr and are still struggling to climb, it's probably not mechanics that are holding you back. It's probably either mindset or your ability (or willingness) to learn. Both are easy enough to fix, the former probably being a bit harder. The first thing you need to accept is that sometimes it doesn't matter how you play, the game will be won or lost regardless of your performance. There are also games that heavily rely on your performance. People like to throw up the numbers 40, 40, and 20. 20% matters, and the other 2 40%s are where you win or lose based on everyone else. So if you always perform well , you should win 60% of your games (more actually, since you can power through some of the losing 40% if you're that much better than everyone else). A 60% winrate climbs extremely quickly. 53% climbs slowly, but noticeably. So if you accept that you'll climb just for being consistent, even if sometimes you have heartbreaking losses (and focus more on your own performance), you'll improve and climb if you play enough games (which you clearly can).
So the other possible problem would be in learning what you can improve and do better. For laning, this is easy as shit. Learn to improve your CSing, your ability to dodge ganks, your ability to control the wave, your ability to trade, your ability to control the lane during your windows of power, your ability to transition your power to other parts of the map, and so on. Like, people may thing Diamond players are good at CSing, but no. We're fucking garbage at it. Like, hitting 160 at 20, while being pretty good, is pretty bad. The reason is because I know I probably missed 20 in the first 5-7 minutes, which is why it's so low. If you watch Challenger players CS, it's actually disturbing how good it is. Like, the odds of them missing 1 out of the first 3 waves (in a relatively even early lane) is the same odds as me missing half of the first 3 waves. But hey, after the first 7 or 8 minutes, I miss a lot less. So I actively look to improve my CS in the first 5 minutes on every champion I play. I don't know what the equivalences would be for junglers, so you'd have to figure that out on your own or ask a high level jungle main. Or a high level player that plays jungle reasonably often. Because honestly, with 1k games of Udyr, you're probably where you need to be mechanically (or mostly there) if you want to get to diamond.