It's only partly about the source, and mostly the fact that hosting and operating a web site is something developers and other IT people can do and don't mind doing.
"Is there anything anywhere that runs this way?"
Co-ops exist at all levels all over the place, and even outperform traditional commercial organizations so, yes.
A co-op is quite a different thing to a loose knit group of volunteers. I’d love to see co-ops picking up this sort of contract, but I do think it’s important people get paid for that work, and the contract is assigned to a specific organisation, otherwise you will eventually end up with a bunch of burnt out people keeping national infrastructure running for free.
I didn't see any such proposal that explicitly described a lack of organization. I assumed the actual implimemtation details were simply handwaved in a casual high level comment. Of course there would have to be some sort of structure.
"Is there anything anywhere that runs this way?"
Co-ops exist at all levels all over the place, and even outperform traditional commercial organizations so, yes.