Now that is an idea that I quite like! It doesn't punish people who dodge because they can't / no longer want to play that game, but someone who dodges whenever they get melee can either play through like the rest of us do or stop playing. (And of course, I do believe that a lot of times you can have fun with a champion you didn't expect to enjoy. Part of why I love the mode.)
Did you just think of this, or have you seen the idea before? I honestly think it'd be worth seeking out Red attention if it's likely they haven't seen it.
A few minor things: what if multiple people are locked to the same champion? It might be possible to keep them from being put in the same game, but even if it is possible it isn't a super elegant solution.
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When a game is dodged, should everyone be locked to their champions or just the person who quit? If everyone is locked, do spent re-rolls still count?**
I think a lot of the frustration in dodging comes from those "I finally got a champ I wanted and then someone dodged" moments, so it could alleviate that if everyone was champion-locked. Of course statistically speaking dodges ought to save you from champions you don't want to play at roughly the same rate, but I think negativity bias might come into play here to make dodging feel worse than locked champions.
On the other hand, I almost always burn my rerolls to try to get something the team needs, and it'd be kind of frustrating if I, for example, rerolled to get an ADC on an all AP team only to have someone dodge and end up on an all AD team with no rerolls.
Maybe I'm overthinking this though, If the system actually got people to stop dodging because they didn't like their champion, dodging should happen way less often anyway. Maybe so rarely that it doesn't matter if everyone else keeps the same champion or not.