The policy is general. MO is mentioned only because it was the only site where this happened regularly with at the time of the discussion. In practice the policy has applied to all other sites including CS.SE.
The policy is about what we do on cstheory. Other sites are free to act as they see fit, e.g. they can close their copy of a cross-posted question if they want to. We on cstheory don't do so as long as the requirements in the policy are satisfied, and I don't see a reason to forbid cross-posting as long as the requirements in the policy are satisfied. Note that it is not just one week delay, the OP also accepts the responsibility to keep both copies up-to-date. The benefits are: 1. other cstheory users interested in the question who do not visit other sites will get any new information about the question, and a2. cstheory users can apost new answeranswers if they want to.
I think the policy has worked reasonably well here on cstheory where users are usually keen to share what they find out about their question with the rest of us, and any improper uses of cross-posting have been dealt with well using the current policy. Whether the same policy would work on other sites I don't know.