They likely already know where a lot of their failures are going to be even without tests. The effort involved in fixing those known failures alone is prohibitive. The fact that the rest of regression testing will be easier with tests is nice, but doesn't reduce the work required to a level where they can justify doing it.

Now you can (at least partially) avoid writing new code that wouldn't work on 3,

Once you're on 3, you can just run your code and see if it works on 3. You don't need tests to avoid writing code that works on 2 but not 3.

as well as gain the option of refactoring things to resolve compatibility - with lots of freedom to decide how much effort to spend on that if any.

I don't think resolving compatibility issues is optional...

It is if you stay on 2 for production/qa and the bulk of new feature development in the meantime. Only CI (and any developers working on python 3 compat.) would be running 3 and only to provide reports of how compatible things are.

Only once the test suite passes would you swap QA to 3, start removing 2 compatibility code (i.e. whatever you've done to hack around handling strings) and eventually ship 3 to production.

Edit: The solution of "just switch to 3 then make it work" is exactly what they can't do because of (completely valid) business and political concerns. I think that if they were able to have a goal of some small percent of test progress on Python 3 per release - but continue to ship features - they would be able to find time to do the work if it's at all a priority, even if it took many many releases. If it's not at all a priority then maybe that's the one-sentence summary instead.