Same code == less reliable independent verification. So open code is good but independent verifiers should try to reimplement the software needed to verify an experiment.
Totally agree, and I'll add that in my experience non-scientists frequently conflate "reproducible" with "verifiable". Simply downloading a data set and code, and rerunning it, is not really a scientific endeavor. To verify a scientific conclusion, other scientists need to design and run their own, independent experiments aimed at testing the same hypothesis. That said, it seems true that open source code can go a long way toward making that possible by reducing ambiguity about what, exactly, was being tested, and how.