First, there is a question of theoretical vs. practical reproducibility: true, in _theory_, an open-source codebase with many millions of lines of code of highly complex transformation can be checked for accuracy by anyone. In _practice_, only very few insiders will have both technical understanding and time to verify the correctness of this or that algorithm, and everyone else would need to rely on their expertise. That situation is no different from using a proprietary system if we trust their authors.
Second, while it might be shocking for someone, most of the science, outside of maybe math, it's not _practically_ reproducible. Most of the articles are behind paywalls, don't have enough data to replicate their results (even economists much more often than not won't include their raw data or algorithms or both). Also, no one who is trying to build a scientific career would try to replicate someone else's results, especially if it requires some costly equipment, reagents, etc. - the rewards in the scientific community are for novel results. Replication crisis[1] describes this pretty well.
Most of the software that runs LHC at CERN is probably not inspectable by a regular Joe the physicist, yet somehow the physicist community somehow trusts their results, etc.
In summary, while an open-source software of Mathematica quality would be awesome in theory, I highly doubt it would be practical for a long time.