So I joined this company as a lead of a team of 4 developers. I found a medium size Java codebase with not a single test. Instead of unit tests, we had a list of scenarios, and at the end of each sprint a QA guy would manually go through each scenario and verify the software.

One of my first moves was to ask the developers to write tests for their code. I got terrible pushback, specifically from one of them that said the dreaded "we don't have time to write tests".

This went back and forth a few times until it got escalated to the lead of the whole project, who sided with the developer: writing tests takes too much time and we don't have enough.

Two days. It took me two days of my own time to go through a completely unfamiliar codebase and get a reasonable code coverage with unit tests.

The benefits were immense and immediate. For starters, we caught a bunch of regressions before a new release that might have taken days of manual testing by QA. We were also able to ship faster and with more confidence.

As OP say, if you don't have time to test, you certainly don't have time not to test.