This has been reported elsewhere, from what I gather, it is due to sandboxing in the OS/app. I don't think we can repackage this as a PWA/web app type thing at this time due to the dependencies and tech debt it would pull in. I don't believe it is to do with cookies, we already use the browser's internal storage. I believe the Userscripts/Hermit/PWA type extensions don't support loading remote resources fully when flattened into a homescreen app due to some security consideration. I haven't investigated it thoroughly yet, but off the cuff, I'd venture to guess it's intractable at this time. It is a minor inconvenience, but I'd suggest just bookmarking in your browser for now. I'm not a mobile ecosystem developer, so I can't speak too much to how, particularly iOS, sandbox applications. I'd venture to guess that when you "bookmark" something to your home screen, it is not really a bookmark, but rather a sandboxed instance of Safari, so it can't see through to the scripts located in the filesystem.
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