Let's do it!
One of the biggest advantages of that idea is that dupe suggestions start visibly competing with answers. So when a dupe gets asked, people actually notice that there may be a perfectly good Q&A already, instead of mindlessly answering it for the 30,000th time - which leads to a heap of mediocre-quality dupes with no connection to each other.
Another advantage of providing dupe-votes as answers is that the user - often enough a newbie - gets an actual response that they can vote on, and accept. They can also react directly if they think the closing is in error (by commenting underneath the answer).
Now, Shog's suggestion, as it stands, would probably entail major changes to how the system works.
How about implementing a "lite" version of it - one that would require hardly any change to the system.
Every time a user votes to close a question as a duplicate of another, offer the automatic, optional creation of a CW answer with, say, the following message:
You can automatically generate an answer that contains the duplicate link. You can edit it to provide the question's author further guidance while still pointing out that the question is a duplicate. You will not gain reputation for this answer.
If the user agrees, the auto-created answer would look like this:
This question may have a duplicate here: How to Flagellate the Blozbaz
It can be edited by the OP in order to provide guidance specific to the OP's situation. This works fine in cases like this one (which is in no way special, it happens all the time around the site. Just an example from 10 minutes ago)
It can be voted upon, reflecting the community's agreement or disagreement.
It can be commented upon, allowing for discourse about whether the dupe-vote is correct or not.
The only problem that I see with this is that users could mistakenly end up believing that an upvote on the answer means an automatic concurring vote, which it doesn't. That would require a message shown to 3k+'ers saying something like "Click 'close' to confirm". It's not a dealbreaker IMO if the idea is otherwise good.
@CasperOne's criticism on the original suggestion about this possibly encouraging link-only answers is well taken, and of course he speaks with some authority as a moderator who has to clean up the damn things. Still, I think the overall benefits would outweigh the necessary extra education.