Eh, I'm not so sure about the funding side there, researchers are not really caught at all and are fully responsible, IMHO. Peer reviewers exist to enforce community standards, and are not influenced to avoid reproducibility concerns by funding sources. The results are always more interesting than reproducibility, of course, and I think that's why the get the attention! Also, there needs to be greater involvement of grad students (who do most of the actual work) in peer review, IMHO, because most PIs spend their day in meetings reviewing results, setting directions, writing grants, and have little time for actual lab work, and are thus disconnected from it.

There needs to be more public naming and shaming in science social media and in conference talks, but especially when there are social gatherings at conferences and people are able to gossip. There was a bit of this with Google's various papers, as they got away with figurative murder on lack of reproducibility for commercial purposes. But eventually Google did share more.

Most journals have standards for depositing expensive datasets, but that's a clear yes/no answer. Reproducibility is a very subjective question in comparison to data deposition, and must be subjectively evaluated by peer reviewers. I'd like to see more peer review guidelines with explicit check boxes for various aspects of reproducibility.

I think publishers care about this a lot, but most researchers do not seem to care as much about reproducibility.