I would love for Champions to have more personal in-game interaction with each other. Vi calling Caitlyn "Cupcake" and the Alters in Twisted Treeline talking to the Shadow Isle Champions are examples that I would like to see more often. That said, cutting out the Bio section entirely... Absolutely not.
For the past week now I've been reading Dota 2 Hero Bios. They are terrible. Their most recent Hero's entire Bio consists of "He was in a big bar fight. He won. When everyone got up for round two they dared him to go fight in the war". His entire Bio can be summed up as the last two sentences from Gragas' own Bio; he walks into a bar, fights, gets recognized as an awesome fight, goes to join the big fight scene that is the game's setting.
Another thing is that the Champion Bios are one of the few things that are keeping the narrative of Runeterra moving. A Noxian Assassin just tried to kill King Jarvin III in Quinn's bio. The fact that he failed is of little importance to my point; the fact that the event happened at all and that we learned about it in her bio is important though. Yes, IronStylus's "Demacia Needs Heroes" topic which covered the same event in considerably more detail was better at conveying the weight of the event that it happened, which is a good thing and I would like to see again, but those kinds of topics are not currently common place.
If the bio was removed, and there was no sneak peak topic like "Demacia Needs Heroes", then there would be no narrative movement to the world of Runeterra. And, to be quite frank, after reading 2/3rds of the Dota 2 Lore and reading the JoJ as well as a number of LoL Lore videos made by TeamTargran, it is quite obvious to me that there is a substantial difference in the story telling between the two games. Dota 2 has better character interaction in-game. I would like to see the same kind of interaction between the LoL Champions in-game. But outside of the matches, Dota 2 is a static world. Runeterra has had a ton of events that shaped it that are told to us through the bios.
Even if you just take the Bios alone, forget the Judgements, the JoJ, the AMA threads, and anything else, you can still get a look at not only the bare basics of the Champion's attitude, but also a bit of the culture of where they are from. Since Runeterra is such a small world there is a lot of overlap since many champions will end up coming from the same place. Once you cross reference the champion's lore you can get a better idea of what the general area of the world is like there and there by get a better understanding of everyone who comes from there. The lack of this is one of the reasons Nami's Bio was slammed so hard. She just pops up out of nowhere, from a place that is completely removed from everything else, and the only thing that makes her relevant in terms of Lore-wise is a tangent line that means that her people occasionally interact with the Lunari... that is the only connection she has with the rest of the world outside of the fact that she fights on the Fields of Justice. Fizz would be just as bad except for the fact that he immigrated into Bilgewater and has adopted that culture for himself thereby getting common ground with the other Bilgewater Champions and by extension Runeterra as a whole.
If you take out the Bios you lose not only a major asset in telling who the Champion is, but you also cutting off a method of world building.
@Everyone saying Riot doesn't care about the Lore: Go read Dota 2 Lore. Almost every single Hero bio reads like Nami's lore. THAT is what not caring about the Lore is like. Riot puts more than a token effort into the Lore and it shows. I can agree with wanting more, but I will not say that they don't care at all.
Sorry everyone. I didn't want to rant, although I usually do, and this was something that I wanted to say for a while now.