On the private location, there are two components installed: Active Gate and the Synthetic engine. Both are needed to execute HTTP and Browser monitors.
The message means the Engine's process is not alive. It is guarded by a watchdog and should be always up and running.
Please create a support ticket for that.
On windows, the engine's service name is in fact "Dynatrace Synthetic" - the one below "Dynatrace Remote Plugin Module".
It must have been stopped at the time the message "Synthetic Engine assigned to location XYZ is offline. Verify its configuration" was being displayed.
I've never heard that those two can't come together so perhaps disabling the plugin service triggered some action on the engine's service. We could make it clear after a detailed investigation.