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I also can't get it to stop suggesting experiments from that data source, or even get it to show at the bottom of the list

Yeah, archiving is definitely something we can add support for. One slightly hacky workaround is to make a new project called "Old Data" or something like that. We recently added the ability to restrict a data source to a specific project, so if you do that it won't show up as an option throughout the app unless you specifically select that project in the left nav.