As a JOSS reviewer and author of papers in academic journals, I’d like to offer my perspective. Please keep in mind I am in industry now, so there’s a bias there.

The first thing to point out is that JOSS has standards for software: tests, documentation, functionality, to name some of them. Reality is that academic journals do not care or ask for any standards and that’s why you end up finding crappy code in many of those repos. Code that is essentially useless because is not tested, not documented, and may be even irrelevant. (All of this you already mentioned)

Then the questions are: what do you want the publication for? To demonstrate skills? To simply get citations? To be a good citizen of the world and share the code? What is what you want to achieve?

IMO, If you are in industry, JOSS is way more valuable. Unless you are doing basic research in industry, your academic publications will have no impact in your career. If you would like to eventually join an open source project as a core developer (to give an example) or to apply to a company that is aligned with the open source philosophy/values, JOSS will be more interesting than other purely academic journals.

JOSS and the open source foundation have an ideal in the background: to make shit reproducible, even if it takes time. And partially, JOSS would like to change mindsets like the one your supervisor seems to have. Because it is by sharing the code that science and technology progress. We want to get out of that stupid loop of “in house software” used ten thousand times to publish ten different papers in one single research group. If the research is computational, then the code must be peer reviewed as well, and that’s what JOSS offers.

Just out of curiosity, have you submitted to Nature? Have you seen their form? I’m asking because even monsters as Nature or Science do not require code or data. It’s “optional”.