Whenever somebody says these things, you know they have not taken any elementary statistics course

Trash argument #1 - Elo doesn't estimate skill.

I agree, if winning games doesn't take skill.

Trash argument #2 - Win rate means nothing when considering a power level of a champion.

This is basically an extension of trash argument #1. I agree only if you don't care about winning and losing games.

Trash argument #3 - Some champions have high win rates because only mains play them.

If you looked at the data you would see there is no correlation between win rates and pick rates.

Trash argument #4 - Some champions have low win rates because they get overplayed.

Basically the same thing as trash argument #3.

Trash argument #5 - I can make a better rating system than ELO by considering in-game factors.

There is NO rating system devised in any game that attempts to measure skill by looking at in-game factors. This is because it is automatically not the UMVUE. Estimating skill based on wins and losses is far more efficient (and accurate) than looking at KDA, CS, or wards placed. Every rating system that is currently used is ELO or an extension of ELO (other factors are NOT in-game factors).

Trash argument #6 - Some champions take more or less skill, which affects their win rate.

Champions generally have the around the same win rate no matter what rank you are looking at. Trash Sin still has a 46% win rate in challenger, and he is probably the most mechanically challenging champion in the game atm. Same thing with "no skill" annie, basically the same win rate in all ranks. The problem is that league is so brain-dead easy that skill does not affect the power level of the champion.

Argument #7 - Some champions have a high win rate because people don't know how to counter it.

This is actually a legitimate argument. People are trash and don't know how to counter things. Having a 60% win rate can still mean the champion is balanced if there are enough counters that go positive against this 60% win rate champion (and the champion has a 60% win rate only because people are trash and do not counter it). However, there are some champions that have no counters (or too little counters that get banned out). These champions always win/go even against any other pick. A well balanced game would be like nontransitive dice. There are some dices that do well against many other dices but as long as there are enough ("underpowered") dices that beats those strong dice, then it is still balanced. The problem is that there are no dice that can beat those strong dice, which is why balance is such a joke in league. Currently league is about picking the strongest dices and not being worried about getting counter picked. Example, Ahri is actually a 54% win rate overpowered champion because she has no/little counter picks and goes even/positive against every other mid lane pick.