This is the sort of situation where I'll consider progressive testing initially - i.e. write out the existing warnings to a file that you commit, add a test tha

This is the sort of situation where I'll consider progressive testing initially - i.e. write out the existing warnings to a file that you commit, add a test that fails if you get any that aren't in the file. As you fix the inherited ones you can regenerate the file, hopefully smaller each time. "If I don't have time to fix all of this -now-, I can at least make sure it can't get any -worse- in the mean time" is a very useful approach when automating the 'make sure' is something quick enough that you can find time for that.