Dwight's idea of using the start date as a soft deadline might be workable for you. In my case, though, I actually use the start date as a start date for certain kinds of projects/tasks, so I think it would probably just introduce additional confusion.
My usual way of dealing with soft deadlines vs hard deadlines is to set the soft deadline as the due date, but include information about the hard deadline in the notes for the task, and tag tasks where the due date is a soft deadline with a context of @SoftDeadline. As I do my weekly review and plan for next week, I have a view which looks at all tasks with a context of @SoftDeadline for the next two weeks and assess whether the current deadline is achievable or needs to shift, then adjust accordingly. You could adjust the timeframe of your view to suit your usual style, of course.