This primarily went wrong because I had the "Use JWT Authentication" checkbox checked...

Thanks @Tani Frankel! I suspect the Support guidance is for when it is used in combination with HealthShare ODS production instances. When you run a local IRIS FHIR Repository as a developer, I believe there is no harm in using this. So the trick I was looking for is:

set ^HS.FHIRServer("dev") = 1

After refreshing the fhirconfig page, the new tile pops up: