The biggest problem with this game...
Click bait title, I know, but I have an honest point to make.
Said biggest problem with league right now, from my perspective is Two Fold. League's utter lack of consistency and it's horrible new player experience.
Lack of Consistency
I don't mean consistency of balance, or champ selection. Nor do I think the game is too buggy or that certain champ(s) are breaking the game. By consistency, I mean that the very way you play the game always changes.
Not to be mistaken for variety, variance or "ways to play." Changes to visual style or minor improvements such as graphical, or mechanical clarity( kill indicators for example). Rather, its the drastic changes to already established mechanics, and features.
To put it in layman terms; if you go off right now for a break and come back a season or 2 later, then its fully possible that much of what you know would be worthless. League has a steep learning curve and a meta that is so vicious that rek'sai would be tame compared to it. People lapse out of games from time to time; comes with the territory of a game that includes 1 main mode, and where much of the variety comes from the champs played against and with.
Reworking champs to be NOTHING like their original incarnations is a problem. Galio is the go to boy. With the exception of his w, nothing in his kit is remotely like the old galio. Irelia, Urgot, and Yorick come to mind as well. Of course, you could point out their low winrates/playrates or their inherent balance issues with their original incarnations, and that is worth considering, but my point is the lack of consistency, not balance. If I leave the game for a season and my main was brand, and I come back to find that he is an entirely different champion, or if I was a mundo main and the jungle camps where changed to make my current clear paths worthless and require entirely new ones that mundo might not even be able to do. I might not bother coming back. I'd make a stupid post on this board (Like I am right now...) and then never come back. It doesn't help that riot (for some reason) doesn't hold back-ups of prior code, making the process of undoing a remake cumbersome and unlikely to happen.
Its petty, but possible and I KNOW that people have quit because of minor things like this. League is terrible at bringing back lapsed players. Which is a problem because it an't doing so hot with the newbies either.
New Player Experience
Most games limit the players options so that they could become familiar with the individual systems independently and slowly open up to allow a gradual transition. When you start this game, however, you are limited in all the wrong ways. Summoner spells, Runes, and gamemodes These things are limited in what you can have base on account level. On paper this makes sense, but in practice it is STUPID. As these are the simplest things to understand. Their descriptions are clear and what uses they provide are stated outright. Its the in-game system that need tutorializing and this game falters on that.
Firstly, nothing is explained in-game beyond the basics. So for a newbie, the recommended tab is all they have to go on and the descriptions are vague and don't really explain why tear is good on ryze or why B.F.Sword is bad on Maokai. Granted, such explicit detail would be cumbersome in a game, but the in-game tutorial is fairly basic and covers the general game only. Tutorial videos on each champ would be a step in the right direction.
Right now, when I click on a champ in my collection, I get a general description of their backstory and a chart, but no idicaton of what that chart means. Since a lot on the chart and stats are subjective and even the ones that aren't don't indicate what difference they have. For example, 
amd 
have the same Damage, but the former is considered easier and has more Mobility. Based on the chart alone, Draven is either equal or inferior to lucian and is harder to play to boot. Besides a basic description of each ability, there is nothing to really help players to learn said characters.
Smart people would note that Riot has a Youtube account with just the kind of videos I'm asking for, but they don't link to them directly on the champ. Instead they sit deep within their youtube account, buried in playlists separated by year. Not every champion has one, and some are outdated. (Ezreal is a good example since its REALLY outdated, but since ez hasn't been rework, they don't need to do him again.) You'd think the learn more button on the champ page would help but it links to the lore page. Nothing that would help improve play.
Itemization, team comp, build paths, clear paths, freezing lanes, the list goes on and on.
Thankfully, the community has provided very detailed tutorials, but that same community is also toxic at times and could easily drive away noobs all on their own. Even then, its poor form to consider a problem fixed when riot never links to any of these sites either through their client or website. I pity the player that doesn't know of op.gg and thus gets into a disadvantage from the start since many apps use it and things similar to give helpful personalized advice.
Add in a terribly long grind to get 20 champs and to level 30 and you have a very slow grindy game people might not want to play.
I love this game, but it isn't the same game I came to love. There are moments of similarity, enough moments that I can fool myself into thinking nothing has changed, but they come less and less. I love the laning experience and slowly that's being taken away from me. Not yet it has, but give it 2 or 3 seasons and it might. At this point, a smarter person would have made suggestions on improvements, but I am not that man. I don't have the full picture, and its quite likely that even the small part I do see I could be seeing wrong.
And personally, I hope I am wrong. And that league's biggest problem is something else...Like Yasuo or something.