at least in some parts of computer science the solution is easy: do not ever publish results without the public source code of all the experiments.
In most cases, peer reviewers will just assume that authors claiming the "code is available" means that a) it is reproducible and b) it is actually there.
As a counter example, this recent splashy paper
https://www.nature.com/articles/s41587-021-00907-6
claims the code is available on github, but the github version ( https://github.com/jameswweis/delphi ) contains the actual model only as a Pickle file, and contains no data or featurization.
So clearly, the peer reviewers didn't look at it.
I think it's more important for reviewers to read the source, the same way one would read an experimental protocol and supplementary information, mainly checking for discrepancies between what the paper claims is happening and what is actually being done. In the above example, a reviewer reading the code would have spotted that the model isn't there at all, even though it runs fine.
In fact, I'd actually go further and question what kinds of errors could possibly be caught be running the same software that the authors did? Any accidental bugs will remain, and any malicious tampering with the experiment data is exceedingly unlikely to be caught even with a careful audit of the code.