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.

Frankly, I was pretty blown away by how poorly it's understood by physicists (although they are probably better than biologist). Even in experimental particle physics, where statistics are essential, the statistical techniques are all self-taught and seat-of-the-pants. I am confident everything being released by the major particle physics experiments is statistically sound after it goes through the collaboration review process, but there are certainly poor inferences being made at the level of the individual physicist. I listened to at least a couple of confused conversations between smart particle physicists about whether frequentism or Bayesianism was "right".

I think it doesn't have to be just "more stats classes". They also need to be focused on the kind of work they'll need to do with those statistics. If you teach future biologists or medical doctors "so there is this theorem that says that if there is statistical significance of A and not of B, then the value of P equals the..." they won't learn it/use it.