It does exactly what you originally asked for. Then you added another requirement involving a restart. No trigger survives a restart. In fact, neither does the value of a sensor’s last_changed property. On startup, it’s set to the current time and date.
You may wish to consider creating a Trigger-based Template Sensor that uses a State Trigger to monitor the printer’s state-changes (from/to idle) and reports the time difference (in seconds). Your automation can use a Template Trigger to detect when the value of the Trigger-based Template Sensor exceeds 600 seconds.