First, your mentality is amazing OP and I hope more people adopt it. Second, this is a huge wall of text, so be prepared.
I'm not sure about all the others, but statistically one can argue Vayne and Ahri are currently too strong because their pickrate is higher than most other champions of their class, accompanying a notably higher than average winrate. Ahri has an ~53% winrate and ~18% pickrate in Plat YoloQ. Her TeamQ stats are similar, though more sporadic. Vayne has an ~51-2% Winrate with a 37-8% pickrate in Plat YoloQ. Like Ahri, her stats are similar, yet more sporadic in TeamQ. In Platinum and Diamond, both are in the Top 10 picked champions this past week, since the patch. Both have positive W/L ratios as well, whereas others in their classes have negative W/L ratios (which is by all means healthier because a popular champion does not equate to an imbalanced one).
While these statistics alone don't prove anything on their own (tho Vayne's W/P stats are ridiculous for any champion), when you compare them to the majority of other champions in their roles, you begin to see that the statistics have weight.
Most other mages right now either have higher winrates with lower pickrates (Veigar, Malzahar, Donger), mild winrates with lower pickrates (Vel'Koz, Brand), or low winrates with low pickrates (Viktor, Orianna). I was trying to find a good contrast (a champion seen often mid, with a decent pickrate and winrate), but since Viktor's/Orianna's recent nerfs that has become kind of difficult. Only Lux compares to Ahri in terms of W/P ratios, but Lux was also buffed recently (so all improvements since then were buff-related) whereas Ahri was nerfed slightly. Ahri is also in the middle of the Top 10 highest winrates in the game, where most of the other champions around her have half the pickrate she does aside from Lux and Elise, who were recently buffed a good bit contrasting Ahri's recent nerf. Veigar has a higher (57%) winrate, but his pickrate is half of Ahri's right now.
For a comparable bottom lane reference for Vayne, let's look at Tristana. She has only recently become popular due to the meta and sits at a ~46-47% winrate with a 26% pickrate. She is also in the top 10 across the board, but her W/L ratio is significantly less than Vayne's, who has a ~10% higher pickrate. I think Tristana is a decent representation of a champion becoming a reliable meta pick, whereas Vayne has become more or less a monster that has excessive power in multiple places to give her truly the leading edge. Vayne is also the second highest champion picked in the game at every level, except Masters where she is third. Again, this information alone is not significant until we compare with other champion in her role. Jinx can also be used comparably, but the results aren't that different than Tristana's albeit her Winrate is ~4% higher. That puts her closer to Vayne's level of balance when looking at it closer because she has a positive W/L ratio to accompany her high pickrate. The next highest contender is Caitlin, with a modest 50% winrate and 16% pickrate, though her pickrate drops to 12% in TeamQ.
I mostly want to point out that while certain champions are not necessarily overpowered and one simply needs to readjust their strategies to defeat them, that does not exclude some of those same champion from having statistical trends leading otherwise. I'm probably going to be downvoted for arguing against a great mentality (seriously, everyone should look for solutions to win over blaming balance) and also suggesting Ahri's being OP, but that's the cost of posting a genuinely strong belief backed by statistics that some of these champions are genuinely too strong for the health of this game.