I've attached an image, (sorry if I did it wrong, I'm more a lurker than a poster) with what I see are some very basic edits to ranked/draft pick that would alleviate some very real concerns about team composition coordination.
1 The ability to choose champions is now enabled similar to blind pick, with a 39% opacity darkening placed over champions picked by summoners not yet up for picking champions.
1.5 Champions picked this way if banned or already picked will populate with an exclamation point and can't be picked denied symbol. See Warwick bot picking jax when he is already banned.
2 The ability to honor, report, add friend, and chat are now available from the menu.
My arguments for #1 are as follows. A picture is worth a thousand words. Sometimes it's easier to click through a list of champions and show (only your team) what your support options are etc.
Some people don't speak english well, or (for whatever reason) refuse to type into the team chat box. This allows you to pick around them if you're willing to just fill spots in your team and not leave you with a "2 mid or feed" bottom pick.
My arguments for the addition of the report/honor/add friend/personal chat are as follows: Add friend: Say you want to add them for whatever reason, not totally needed. Personal chat: Want to invite your support with you to skype, but not broadcast your username it to the entire team? This is your ally. Honor: It's my belief that you should be able to honor someone for good sportsmanship for filling a role, or giving up a spot to someone trolling for only one position. We all have seen these people, and we all want to thank them, but 50 mins later we may forget. The ability to honor once more would of course be disabled. Report: The current system is set up to que dodge and send in a support ticket if you are stuck with trolls in ranked/draft. Adding this to the champion select would avert this easily exploited vote to kick people keep bringing up, but would allow you to properly report people for causing dodges. Reporting would flag the entire game to tribunal, and in a perfect world, show the chat logs as well.
Edit from reddit: the ability to honor could just post a popup postgame that would remind you something to the effect of "You wanted to honor this summoner X at champion select for teamwork etc" Which would allow you to honor them if they deserved it, or not if they didn't. And it would eliminate them needing to make a new honor category, or allow people to get honor que dodging.
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I agree with your idea 100% where you can report from character select, you might want to re structure this tho it is hard to read.