Will LoL ever reach Dota levels of balance where almost all of the champions are picked in competitive?

I don't know Dota nearly as well as I know League, so my answer may reveal a certain amount of ignorance, but I'll be interested in seeing the discussion that ensues.
My understanding of Dota is that counterpicking at champion select has a much stronger influence than it does in League. In League, there is a certain level of counterpicking, especially at the higher elo ranges, but overall we don't want you to lose at champ select. We want how you perform on the map to really determine who wins.
In Dota, again from my point of view, you can lose the game quite easily at champ select, if the other team makes a pick that strongly counters yours. This does mean that diversity can be larger because there are more niches in an ecological sense.
Why am I talking about ecology? In the natural world, you can only achieve species diversity when there are more niches for an organism to live, or else one just out competes the other. Environments like coral reefs or rain forests have an enormous diversity of species because they have an enormous number of ecological niches. Environments like the Arctic ocean have fewer niches and therefore lower species diversity. The same is true of champion diversity: more niches = more species.
A generalist champ with good stats in Dota isn't a safe pick the way it might be in League, because a strong counter can nullify those good stats. By contrast, a champ with limited utility still has a space if that utility is countering another champion. The downside of that is you may then need to play a 30 minute game only to discover that you made the wrong comp choice a long time ago.
Our philosophy for League is that player agency outside of champion select is important, and champ diversity is also important. Yes, we are making life more difficult for ourselves by valuing both things. I also don't mean to imply that one approach is more correct than the other. They are just different games and they too have different niches.