After a little more searching, this seems to be another instance of the MTU issue.
I lowered the network-wide MTU (to 1280) per instructions in Make MTU configurable · Issue #74 · zerotier/ZeroTierOne · GitHub and I stopped having connection issues with my HTTPS service. I presume my 56 byte ping and HTTP services worked because the packets were smaller.
Does ZeroTier have any documentation about MTU? Ideally we don’t touch MTU at all.