I feel like this would be a nice thing to know. However, it is also super abusable and could lead to making it easier to tilt someone.
Like, say you were mid and banned Annie. Enemy mid wanted to play Annie, so enemy mid picks something else. You now know this isn't what they wanted to play. If you can get into game and rile them up in all chat, you can have an easier lane. Depending on the player, their second choice might not be nearly as good as first choice. If your junger ganks more because of it, instead of having a slightly harder game, they could have a terrible one, just because of your team having the knowledge you banned their first pick.
To me, it falls along the same lines as not showing everyone's pings in the loading screen anymore - it was too easy to see that someone had bad ping and could be ganged up on. Now you can see who might be a little upset about not having the champ they really wanted to play, which could lead to increased jungler attention / more smack talking that could cause them to throw the game, as opposed to not knowing if they had their first pick banned.
Like I said, I like the idea because I have also always wondered if my bans ever actually do anything, but I also think it could lead to more harassment in game. And as we all know, this community doesn't need more fodder to be jerks about.