There's less of an opportunity cost with picking up a new character than there is with picking a new role. Learning a new champion takes some practice, but many of the things you learn while playing your new character are applicable to every character in the same role.
New characters aren't as tough to learn
Although there is a bit of learning curve for a new character, particularly if that new character has combos, most characters don't take that much time to pick up. Once you've reached a certain level of veterancy, new champions should have mostly familiar mechanics. If you've used collision-based linear skillshots before, it's not a surprise to use a new character that has one, even if the specific mechanics are a lot different. For instance, landing rocket grabs is more or less the same skill as dark bindings.
It's pretty rare to use a character that doesn't take some level of adjustment, though. I think the most "easy" to switch to are characters like Graves or Jinx that have no really unique mechanics. I also learned Caitlyn in my first game playing her and carried my team just fine. Overall, ADC/marksman characters tend to be simpler to pick up, even if you're not playing on Summoner's Rift.
Other characters do tend to have a learning curve, and I definitely recommend practicing in bot games before you take them onto the field. When I first picked up Lee Sin, I spent quite a few games running around with a Sightstone hopping to wards before I felt comfortable playing him in PvP games. In general I don't like to play a character unless my basic competency level is at my current level of understanding for that character. Obviously, if I don't know any special tricks for a character, I can't learn much more than basic fundamentals.
Although owning a lot of characters is bad for ARAM, it's also just bad in general. If you never play Diana, the time you spent dinking around with her when you first bought her was essentially a waste. I'm mostly saying this as a self-criticism, since I own Diana but never play her (and probably won't unless she gets buffed).
One could argue that a little time spent goofing around with a character isn't a big deal; after all, League is supposed to be fun. Unfortunately, that time starts to add up when you own 40 or 50 characters. I have literally dozens of characters I almost never play, and I like to think that my roster is extremely slim compared to a typical veteran player. I own a little over 40 champions, and while I understand how to play all of them, that's possibly not so good when I only play a handful in ranked matches (Elise/Evelynn jungle, Zyra support/mid, and a few ADC options). If I'd stuck with fewer characters, my mechanics with them would probably be a lot more polished.
I have a friend who has something like a thousand games with a single character (support Leona). He is really good with that one character -- far better than I am with my much larger roster. My wins are split between all different game modes and characters. Just food for thought.
One big disadvantage to ranked play is counterpicking. It's especially prevalent in solo lanes, but a savvy pair of opponents can counterpick your support in bottom lane too, making it very hard to actually get anything done. If you pick something first, there's a pretty good chance the enemy can counterpick you.
The flip side though is that, with a fairly diverse champion pool, you can counterpick the enemy too. If you are confident that you can beat the enemy Fizz with your Cassiopeia (I'm not saying this is a counterpick, but I know it's hard for Fizz), then ask for the lane and say you know the matchup. It doesn't have to be an explicit counter. If you see Warwick top and you're sure you can win with your toplane Udyr, then go for it.
As a quick aside, this is the only real way to counter Evelynn right now; pick a dueling/invasion jungler and disrupt her route. If you pick a jungler against Evelynn that can't 1v1 her, she will rampage all over your lanes for free.
Chasing the meta
If there is a sin for which I was guilty, it is probably this one. I bought Lee Sin only because he's considered to be one of the best junglers (he was the best at the time I bought him) even though I preferred Evelynn and later Elise and was pretty happy with those characters. I hate playing Lee Sin, and I don't really like his overpowered mechanics. I'm still at my happiest playing Evelynn or Elise. I'm feeling the pull now toward getting Shyvana, even if she also does not play in a style that I like.
It's good to have good characters in your roster. If you were planning to make it to Challenger using only Vayne (with the recent nerfs), that would be a definite uphill battle. She definitely doesn't have what it takes to go the distance, so you will have to rely almost 100% on outplaying your opponent.
If you're a flexible player, chasing the meta gives you generally good matchups. Playing Corki in late Season 3 was a huge advantage, for instance. He was very frustrating to play against and felt very powerful whenever he was used. Even pre-nerf Jinx didn't feel as frustrating as Corki.
The downside is obviously that you end up rotating out of characters when the inevitable nerfs happen. While I don't think Lee Sin will be nerfed (it's been tried quite a few times and he's still strong), most other characters have been dethroned by Riot once they were in the top pick spots. Even if nerfs don't happen, metagame shifts might require that you play someone new. For instance, Karthus didn't magically get worse in Season 3, but assassins got better and their counters didn't. Since Karthus wasn't very good against assassins, it phased him out of the League for a while. Now that we're in the preseason and people are starting to experiment more, he's cropped up a little more often since his counters are rarer.
Chasing the meta does slowly fill out your roster with current and former meta picks. It's rare that a character gets completely nerfed to unviability, so this can pad your roster with a lot of viable counterpicks. Is this better than playing only a small handful of characters no matter what? My overall guess leans toward no. If you have a small champion pool and you're really good with it, skill tends to overcome the counterpicks and bad matchups.
We understand what it's like to climb the skill ladder in League of Legends. The Summoner's Guidebook teaches you the tools you need to get a competitive edge. Whether you're climbing the ranked ladder, playing Draft Dominion, or getting crushed by intermediate bots, every enemy has a weakness. And every Thursday, Patrick Mackey shows how you can improve improve on yours.