Make start time picker same as due date picker and block picking start dates after the due date

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Is your feature request related to a problem? Please describe.
The start time uses a different widget, it seems, than the due date:

Moreover, you can pick a start date that lies after the due date. And a due date that is before the start date. The widgets for both should keep the logic into account that a task should start before it is due 🥇

Describe the solution you'd like
Make the widgets the same.
Let the Due Date widget pick a due date only after the start date (if set).
Limit the start date to before the Due date (if set).

Start by locating the start-time and due-date picker components in the JavaScript frontend and compare their current behavior. Verify the shared widget behavior and date constraints for both fields, including unset dates and dates in the wrong order. Done means both widgets match and prevent invalid start/due-date combinations.