This... is... just an all round really great idea... sums up the problem nicely, proposes a logical and sensible solution that promotes and encourages team play, without directly impacting/impairing the enjoyment of solo play (if that's your thing). Having the extra DU on maps also helps with the difference in builder stats and encourages coordinating as a team, it also will break up the standard builds as well because random teams won't always have the best stats to build those builds, it will encourage creativity, which imo is half the fun. Honestly hope something happens with this idea.
Introducing a separate Multiplayer game mode would be an ideal solution also. Give people the option to play this way, maybe even randomize the map when forming groups, allow people to make connections they can carry over into normal multiplayer if the want, or continue to improve strategies and layouts in the shared DU mode. This way we keep the original multiplayer mode a lot of people like, so people can either fully control their build or carry friends for xp/gear. I think a second game mode would really make both parties happy, because the people complaining here about this idea won't be affected (even thought they'll likely still complain lol).
As far as extra rewards or incentives, I agree there should be some, but not in the sense that this becomes a seemingly mandatory method of play to advance, someone earlier mentioned different pet colors, you could even introduce a different armor/weapon model (just the model, like, shiny armor or a flaming sword) that only drops here, it's a bit of a symbol for players to show they've succeeded in this game mode on high difficulties, but is purely cosmetic as to keep gearing equal in both modes. No one wants to be forced into a mode to succeed, and even though no one IS forced, increased drops/better gear would apply a feeling of pressure to those who really want to push.
Again, I'd just like to say solid 10/10 idea by OP, and as long as it's implemented in a good way I think it would do great things for the game, as well as introduce a new difficulty/challenge outside of the basic difficulty settings.