In my chemical engineering work, I have often private routines I want to test. Pardon me, but this is Fortran code, so I cannot have "complex" class structures to allow different level of testing without complicating the code too much and making it unreadable.

The solution is "stupidly simple". I have at my module level a public subroutine with the name "test_module" which performs the tests of all the private subroutines of the module I want to test (these are usually analytical derivative calculations and stuff like that).

Yes, I have test code in my module, but in practice, after 10+ years of using this pattern, nobody had issues with it.