How should I troubleshoot suspected issues connecting to cloud.hubitat.com/api/xxxxxxx?

I have had an issue very similar to yours on two different C7s during the 2.3.9.x beta. In my case, only inbound connections through the Hubitat cloud would fail (remote admin, maker api or cloud endpoint URLs, some integrations). Like you, other things worked fine.

At first I thought it was due to a platform update, but then realized a simple reboot fixed it. However with the last occurence I discovered a simple cable pull was enough. See also : Cloud connection stops working - #21 by rylatorr

Not sure Victor has tracked it down yet, which is why I tagged him.