"For the longest time, I would NOT allow people to write tests because I thought that culturally, we need to have a culture of shipping fast"

Tests are how you ship fast.

If you have good tests in place you can ship a new feature without fear that it will break some other feature that you haven't manually tested yet.

Exactly. OP seems to have very limited understanding of software development if that fact has eluded him.

The rewrite version of this that has gone best for me is to do it as a strangler, not a reset. Pick one ugly workflow, lock in current behavior with characterization tests, rebuild that slice behind a flag, repeat. You still get to fix the architecture, but you do not throw away years of weird production knowledge.

I think the more specific description would be that "not writing tests allows shipping fast today, writing tests allows shipping fast tomorrow and afterwards".

It wasn't too long ago that I wrote tests for something that was shipped years ago without any automated tests. Figured it was easier doing that than hoping we won't break it.