If an external test passes and an internal test fails, the external test isn't really adding any value, is it? And if the root of your issue is "What if test A doesn't test the right things", doesn't the whole conversation fall apart (because then you have to assume that about every test)?
IME this is a common path most shops take. "We have to write tests in case our other tests don't work." Which is a pretty bloaty and wildly inefficient strategy to "Our tests sometimes don't catch bugs." Write good tests, manage and update them often. Don't write more tests to accommodate other tests being written poorly.
> I mean, running internal tests is cheap.
Depends on your definition of cheap, I guess.
My last job was a gigantic rails app. Over a decade old. There were so many tests that running the entire suite took ~3 hours. That long of a gap between "Pushed code" and "See if it builds" creates a tremendous amount of problems. Context switching is cost. Starting and unstarting work is cost.
I'm much more of the "Just Enough Testing" mindset. Test things that are mission critical and complex enough to warrant tests. Go big on system tests, go small on unit tests. If you can, have a different eng write tests than the eng that wrote the functionality. Throw away tests frequently.