DOIs are just one type of persistent identifier. PURLs (Persistent URLs) are another. We would hope that arXiv IDs would be another too. A DOI simply provides a persistent identifier. Its advantage is that it is widely-used, and broadly adopted within scientific publishing.
You are mistaken to think that URLs to papers published through a publisher are stable. I can recall numerous occasions where the URL to a paper didn't work because the publisher had changed the structure of their website.
Why are DOIs better than URIs? I think this comes down to the fact that a DOI is an identifier that is entirely removed from the structure or layout of the website where the resource is located. URIs often aren't just identifiers of content on the web; they can include information about the filesystem layout for example or other structure of the website. The owner of the website might wish to change the structure of their website to make use of a new technology or as part of a redesign. If they do this, the link from the identifier to the resource is now broken but they have no consistent way to indicate this without maintaining themselves forever redirects from old URIs to new URIs. By separating the identifier from the layout of the web resource the identifier doesn't change if the website or server hosting the resource changes.
Also, in this scenario the DOI wouldn't change but in a revised website layout the URI to the resource would be different or you'd have multiple identifiers relating to the same work.
why would a DOI for that Zenodo repo be superior to a URI for the same Zenodo repo?
If Zenodo changed the layout of their site, the identifier would change and they'd need to maintain a link (a redirect) from the original URI to the new location for all time now.
Also, arXiv specifically has arXiv Article-IDs so that if they reorganise the web interface there is still a persistent identifier of each article (and revision). If there were no additional benefit over the URI, arXiv wouldn't have gone to the trouble of creating their own persistent identifiers. (One presumes cost is [or was] an issue in arXiv not providing DOIs.)
The same concept lies behind all these persistent identifiers, like PURLs, arXiv IDs etc. URIs just provide a way to reference a location not an identifier of that work, even if while referencing the location they act as an identifier. The key thing is persistence.
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