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Follow-up: Since it seems all the code is open-source, is it possible for me to run this service on my own server (or at least in my own AWS setup)?

The lack of a license leaves this very unclear.

GitHub's user agreement allows for the free use of any code posted in a public repo.

No, it doesn't. Standard copyright applies if there is no licence.

GitHub user agreement allows other users to view and make copies of your content on github but not for "free use" in general.