The best and simplest approach IMO would be improving peer review. If it was understood that scientific journals would reject papers with poor statistical analysis you would see changes fairly rapidly.
Nature and the NEJM, and possibly others, assign statistical reviewers. So this is a known problem that some high profile journals are attempting to correct.
Yes, but I think one of the problems is that the reviewers can't sufficiently judge statistics. It is peer-review, after all. I've had some reviewers ask some questions that indicated that they didn't really understand the t-test.