I think you can do it with an automation? You’d want a Status Change automation, and you’d pick the one that reads “When status changes to something, set due date to current datetime”.
Then you just update the italicised bits so it reads “When [Doneness Status Column] changes to Started, set [Start Date Column] to current datetime”.
The only downside of this is if, say, they update Doneness to “Blocked” (or something) and then change it back to “Started” once they resolve their blockage, the start date will then update to the datetime of the most recent time the Doneness status was set to Started. There’s no way to set it and then lock it. So you’d have to ensure your users are using workflows where they don’t step backwards through statuses, only forwards.