Without delving into the pro and cons of this stage I'll just come out and say it, mid-season 2, early season 4 and patch 5.1. Those periods were arguably the best times in terms of balance, at least from my perspective.
Mid-season 2 because that's when pro play really started to shape the meta and we saw a huge variety of champs and builds at the time (I still remember, Toyz, his Ori was sick, also he went shit like D.Blade Ori into certain matchups lmao), and before "League of BC" became an apparent issue.
Early season 4 was great since every role felt interesting and we got Spirit of the Elder Wraith and Lizard, which brought with them Spirit Stone which brought Machete, which more or less fixed the jungle role since it gave junglers an actual first item instead of just going boots, D.blade or Long Sword.
Also because of the fact that it was right before any bullshittery occurred cough Yasuo and Rek'sai cough and Morello and the team had more or less balanced each pick relatively well outside of a few exceptions such as Renekton and Kha'zix.
Patch 5.1 simply because it was right before the removal of DFG, so off-meta AP builds were still possible and moderately successful, and this was also the time when Weedwick was fixed and tank meta wasn't a huge issue. Diversity was still just great, if not better since top lane was beginning to see some positive changes in the right direction.
Pretty much, every patch after 5.2, especially the class updates helped to shape this game into the volatile snowball that it is now. Runes Reforged definitely nailed in the fate of this game, however every patch past 5.1 contributed immensely due to the continuous hate of "tank meta", assassins and eventually bot lane. All of this was because of Riot aggressively pushing what they considered to be "unique" champions and reworks to the forefront, giving previously fine characters who filled certain niches too many frustrating and overloading options to work with, and releasing overloaded champions whose kits never should have seen the light of day.
Sorry, but class updates were a failure.
Also, the game started to decline in quality due to Riot removing items and moving their passives onto another item. Example, Malady's on-hit damage passive is now on Nashor's Tooth. Effectively what this did is that it made items exceptionally overloaded and streamlined since there's no longer any choice since Riot has completely removed all choice entirely in favor of overloading single items with too many different features.
Additionally, they also try to replace items and make them more "healthy". Example, DFG got replaced by Luden's. While Luden's appears to be more healthy it isn't quite as healthy as people say since it still adds more damage to your burst, and the bounces are all entirely random. You heard me, random. That's another thing that came with this strategy, we first saw their promotion of RNG or "pseudo-RNG".
Basically, removing items caused item variety to decline drastically, class updates overloaded champions and newly released champions got continually more broken and overloaded as time went on.
Moreover, the keystone system could also be to blame since instead of raw stats. and the like you got one mastery that's excessively powerful that defines your playstyle to the point of absurdity and a few minor masteries to give you basic stats. Initially they weren't a problem, but as we're clearly seeing now Riot has again aggressively pushed changes through to make keystones entirely defines your pick. This also resulted in them completely sacrificing the old rune system in favor of this system, which they were then forced to change; partially bringing back old runes anyways.
tl;dr Any time before class updates, overloaded releases, removal of DFG (and other items) and Runes Reforged was a prime example of League in its most balanced state.....except for late season 2 and early-mid season 3 (99% Pick/Ban Kassadin with 60%+ win rate, I still remember the horror).
Removal of variety killed this game's balance.