The first rule Riot should respect in order to balance the game in my opinion...
Warning pretty long read You have created categories for champions that define their playstyle, yet a lot of them go way beyond that.
If you can put some clear limitations regarding what a champion and can't do that would help tremendously, people can understand better how to counterplay each type of champion.
Like take this one: Assassins are defined as agile, mostly melees champions that have to put themselves in danger by getting near a priority target and have solid escape tools.
Yet within the same class you have got Zed, who is extremely safe at what he does, Fizz, who is the exact opposite, almost recklessly diving into the opponent team after landing an ULT, you have Ahri and LeBlanc, that are dumbly safe(they are classified as burst mages, but they have the mobility of assassins, which by default makes them assassins).
The fact there are so many well designed assassins and then you have this: assassins poking you from range and chipping down you HP bar safely and KITING, looking for a favorable all in as oppossed to Kha, Fizz, Talon and others, playing like real assassins, wanting to get in your face and kill ya, well we have a problem.
"Artillery Mages are the masters of range, and they leverage that advantage to whittle down their opponents over time from great distances." Taken from LolWikiaPage.
Ziggs does that right? Very good poke mage design, fun to play, telegraphed skillshots.
Zoe, Xerath, Lux, other champions classified as Artillery mages, yeah what a joke. They have potent lockdown abilities, that they follow with a devastating burst combo that 1 shots(in case of Zoe it's no combo, just her Q).
Then you complain there is too much burst when someone who "whittles you down over time" 1 shots you.
Xerath skillshots are almost impossible to dodge without boots, they pierce minions with no damage reduction and have massive hitboxes and it's a lot of burst. Zoe, let's not even start.
And maybe we can address Marksman as well, you know who in particular: "Marksmen are ranged champions whose power almost exclusively revolves around their basic attacks: using their reach to land massive continuous damage from a distance, marksmen are capable of taking down even the toughest of opponents when positioned behind the safety of their team"
Kog'Maw has a slow field and one of the lowest move speeds in the game, Kaisa has a move speed buff, that renders her unable to AA while charging, Lucian and Graves have less mobility with shorter range, yet Vayne gets to have a knockback(hard CC), that doubles as a stun, can tumble every 1 sec during ULT and goes invis more often than Kha'Zix. People complaining about her for a while now, Riot doesn't care.
Riven-skirmisher(defined as champions that lack high end burst and are armed with situational defensive tools that gives them an edge in skirmishes). Unlike Jax E, Fiora W, Master Yi W, Trynda R, Riven is equipped with high end burst and "situational" tools as in mobility(dashes) and hard CC(stun, knock-up). She plays almost like an assassin, but she is more durable, hard CC and more burst than average duelists.
Tahm Kench, someone that is not problematic, rather the opposite, he is a tank, but his CC is unreliable, needs 3 stacks on someone for a stun and suppression and he does a lot of damage, like a juggernaut and he is not that either. Like Hashinshin said, they wanted to make his W the main ability and they forgot to like work on the rest of his kit, so ok, let's just give him damage, outtrade anyone early game.
Let's stop over here, shall we?
Bottomline, among the things people complaing regarding balance is champion design.
Since a lot of problem stem from champions having overloaded kits, making them do more than they were designed to, one step in the right direction is to narrow the differances in style of play between champions belonging in the same category(categories determined by Riot itself).
It's hard to implement as it requires kit reworks for a lot of champion, yet it's how you balance this game, at least from my angle.
What do you think?