I can think of three things that already solve this problem, without the need for adding additional complexity to either the site's user interface or the source code:
We can simply continue to close duplicate questions as quickly as possible, before a bunch of incomplete, bike-shedding, and/or just plain duplicate answers have a chance to be posted. Judging by the number of questions posted here on Meta with users complaining how quickly their question was just closed, it seems to me that this system is working as intended.
Sure, some questions don't get closed as quickly as others. But those are generally the ones in the less-popular tags that get fewer views (and thus have fewer chances for close votes to be cast). But in the less-active tags, it seems that your cited problem of rampant duplication would be much less for reasons of scale alone.
(And if you want to help ensure that duplicates are sought out and closed even more quickly and effectively than they already are, support Pekka's proposal to provide reputation bonuses for it.)
Duplicate questions whose answers contain nothing of added value compared to the "master" question can (and often are) deleted shortly after being closed. All 20k+ users can vote to delete immediately, 10k+ users can vote to delete after 2 days, and moderators can delete questions with an immediate, binding vote.
Again, I think this is already happening, judging by the handful of people that have complained here on Meta about questions being deleted.
If you ever come across a duplicate question that has some really good answers that provide information not already provided in the answers to the duplicate question, you can (and should) flag that question for moderator attention and ask them to merge the two questions. Moderators already have this capability, and the necessary discretion to apply it only when appropriate. This is a great option for when deletion would otherwise be inappropriate.
And finally, as a special bonus argument, consider that some duplication is actually a good thing:
Dr. Strangedupe: Or, How I Learned to Stop Worrying And Love Duplication
TL;DR: It seems to me that the tools already in place allow us to deal effectively with the bad duplicates, prevent their number from continuing to grow, and keep it easy to find the quality content that you need. And that the good duplicates are actually, well, a good thing.