I would say teaching is not that relevant specifically for most tenure-track positions at big research universities. It is absolutely something you need to demonstrate some actual experience and proficiency with if you get a position at a small liberal arts college or a community college, where tuition is basically how they keep the lights on.

I’d also say that even at an R1, teaching volume at an acceptable quality is sometimes rewarded if your college within the university is very undergrad-heavy, because it can be part of how the university apportions funds to departments. So, it wouldn’t matter at a med school, but potentially a little in arts and sciences, though still in distant second to research.

There are also a small but increasing number of tenure track teaching-focused positions at big research universities. These folks typically help design and teach the biggest intro lectures and/or other very time- and labor-intensive courses. There are fewer of these positions than I’d like to see in an ideal world, but not zero.