I'm not sure this is the answer you expected, but I found your question really interesting, so here's my answer :)
I've played League of Legends since June 2014, and my account hit level 30 just after the end of Season 3, so the current system is the only one I've ever known. And I truly think the system is the best League has had since its beggining. I mean just a ranking, with thousands of people before you, is just like Riot Support. And a single number is a bit better, but really, what we got now is the more motivating system.
But, yes, I don't really look at who is in my small ladder (apart when I'm looking for friends, or funny summoners names). I don't care about where I am in the ladder. I just care about the divisions and tier, and my amount of LPs (Am I close to go to promotion ? Could I be relagated if I loose some two-three matches in a row ?). This is because I know the ranks in the small ladder doesn't mean anything. I can move up to 30 places up or down this rank with just one game. And, this small ladder doesn't gather all the people of the server that are in the same division as me. So it's kind of meaningless, at least for me.
But I do think that the currend system is better than the Elo one. I do think that even if we don't use the ranking of the small ladders, it's not the same than the Elo's one. It's BETTER.
Why ? Because you have divisions. Because you have tiers. Even if they aren't accurate (because they do are, as a player can have some luck, or some bad luck, and a lot of others things), they give you a global idea of your skill. Because if even with 100games, you stay in the same division, that's obviously your level (except if you're really really lucky or unlucky). And, more than anything, it gives you a landmark (not sure this is the exact word, but this is the point). You've got a landmark. All those tiers and divisions, they're lankmarks. They give you a general idea about where your're in terms of skills, because you've got something like 5 tiers, each one with 5 divisions + 2 extra tiers.
And this is something the Elo system doesn't do.
And this is something, even without noticing, you surely are looking for.
Because with the Elo system, you'd be trying to got some landmarks. You'd be trying to cut the only number you have. You'd be trying to split it into some group of 100elo, or maybe 200elo. You'd be trying to make some kind of divisions yourself. Let's take an example, you've just won some games, you've just gain 100elo, and passed the 1200mark. And you are going to be happy because you passed this mark. So you would have created a kind of landmark yourself. And the current system provides you these landmarks, in a more reliable way.
I hope you understood my opinion :D (English is not my native language, so :'( )
