As a further update to this I've also now tried all of the known 'kickstart' tricks to get the remote ARD client up and running but to no avail.
It would appear that something in Mountain Lion is breaking ARD 3.6 or there is something weird in the new configuration. I've seen a number of other postings around this with similar symptoms.
All of the ARD remote clients running 3.6 are showing as inactive and sleeping, but I am able to screen share and observe.
Come on Apple, if you've broken something, own up and get it fixed.