Does explicitly naming the package name in the function call have its disadvantages?

Only in that it means you have to do extra typing.
If calling it that way works but calling it as functionOne produces the error you reported, then that's a clear indication that you've managed to disable the exporting of that function.

Btw, the general advice is that you should delete the "&" and call the function as MyModule::functionOne

Cheers,
Rob