So, the boards are "the vocal minority", but how else can they get any feedback from their playerbase?
For things that are purely numerical in nature, such as champions being over/underepowered, they can go to the numbers they have and look at statistics, (scores, win rates, end-game stats, etc.), but that doesn't work for subjective things.
As a random example, I get the impression that people generally dislike or hate Kalista because of her passive. Threads on boards seem to support this. But how can you know if it's just a vocal minority, or if that's a general sentiment?
I suppose you could look at ban rate - Kalista is the 13th-most banned champion, despite having a below-50% win rate. That implies that people hate playing against her, but it could be due to a number of other reasons (people think she's OP, people hate supporting her, etc.)
Another thing they could do is to actually poll players on gameplay issues. I'm not saying they should ask them about balance ("Do you think Ekko is overpowered?") because that would be pointless, but to instead ask about perceived game health ("Do you enjoy playing against Kalista?")
I personally think that these kind of considerations (is this champion fun to play with/against) are more important than raw numerical balance anyway. An overpowered champions is kind of annoying, but a badly-designed one saps the fun out of the whole game.