How can I introspect the "shared subtree" mount flags for a particular mount?
Check /proc/self/mountinfo. For instance, in one of my systemd-based Ubuntu VMs / is shared (systemd enforces it at boot time):
$ cat /proc/self/mountinfo | grep ' / / '
22 0 8:1 / / rw,relatime shared:1 - ext4 /dev/disk/by-uuid/e176bae0-84a5-4bd2-b968-e809e902488c rw,data=ordered