I don’t think that’s true. It should definitely find script.start_spridare and update whenever the state of the script does. Adding entity_id: sensor.time will just make it update way more often than it needs to.
timestamp isn’t a sensor or sensor type, it’s a device class. The updating, I believe, happens in the frontend (browser, whatever.) The entity itself doesn’t have to update its state in the State Machine for the value shown in the browser to update. E.g., I have some entities for solar events (from my custom component, er, integration) that only update once a day, with a device class of timestamp, and the “x hours ago” or “in x hours” display updates as time goes by.