I think the biggest problem facing science right now is the scale at which it is occurring. My field does not have this problem, but I can't imagine what it's like trying to stay on top of the literature in e.g. medical sciences. Paper writing and peer review work at much smaller scales - there are order hundreds scientist in my field and I think it's functioning just fine. For fields with thousands and 10s of thousands of active researchers, I think there needs to be a different way to vet and organize information. I don't know what that is exactly but it might look something like twitter (can't believe I'm writing this!). On second thought, maybe it looks more like HN!
The misalignment of incentives can be problematic as well, but arguably, not the biggest problem.