Tenured mathematicians care about publication. My impression is that this is because they believe in the process of peer review. However they also have other incentives as their publication record affects their ability to attract research grants. They also typically share authorship with untenured students and postdocs who need publications for their own Tenure tracks.

You are right that most work gets published in mid tier journals. These are still peer reviewed. The difference is that they have a lower bar for impact. The bar for correctness is still very very high however.