I like this. On sites like Physics, the community doesn't participate much in deletions (and we don't have enough active high reps anyway). As a mod, I could do it, but none of us have really felt the need for it. This seems to solve the problem.
However, come to think of it, do we really need to delete the posts? The point is to avoid the broken windows problem. Instead of deleting them (making them unsearchable and unreadable), here's a radical idea:
Just make them hard to get to.
Basically, any 7 day old closed post will be hard to get to unless you specifically search for it. If I search for something, I should get a little link at the top that more or less says "There are 30 old, closed questions that have been excluded from this search. [Include all closed questions]". Such questions should also disappear from all question lists (like the "highest votes" list). This cleanly avoids the broken windows problem. A visitor won't ever come across these, and neither will a regular user unless they explicitly search for it. Deleted questions sometimes have good answers. In other cases there are meta discussions about these. There are many times when I want to see a q on SO, and can't -- because the deletion army got there first. Giving them the silent treatment instead of deleting them avoids such annoyances, while still solving the broken windows problem.
I'm not entirely sure of this; but I'd like to put it out there :)