Correct me if I'm wrong, but this new argument seems to work whenever there is a right-invariant metric such that the exponential map is a local diffeomorphism, provided that $V$ is small enough to be a diffeomorphic image of an open ball under exp. I gather this is not known for $Diff_c(R^n)$ but is for Diff(S^1) with the $H^k$ metrics, when k is at least 1?

Thanks! although I'm not sure how simplicity might help here (?) Here's a different way to think of the same kind of question with a more metric formulation: put your favorite (complete, left-invariant) metric on Diff_c(M). Is there k so that every diffeomorphism in the epsilon ball about the identity is the product of k elements in the epsilon/2 ball?