The main thing with TDD from my understanding is that tests are the requirements, so anything that falls outside of the tests is by definition irrelevant. Most of the "test everything" recommendations come from the TDD mindset, so if you try to apply that outside of the TDD framework it can get messy.
This perspective helps limit the scope and coupling of your tests, since there is typically an astronomical number of tests that you could do, but a very finite number of testable requirements. Refactoring should not generally break tests, but if refactoring occurs across/between several modules then you will probably have some rework, but I would argue that that is more of a "redesign" than a "refactor".
One good reason to test every module/class is to reduce the scope of any bugs you do come across. If I have a suite of tests that demonstrate my module's behavior then I know where not to look for the bug. With integration/system tests alone you will have some searching to do.
I always have the feeling that is still a problem.
I get rather high leven requirements, but they are implemented by many parts of the code. So simply writing a "Req1 passes" would require to implement many many thigns till the requirement is met.
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