You can only specify a date with checklist items, but not a time as well. They are considered to be too granular to be time sensitive.
The calendar displays all the cards visible on the board, so you could add an automation rule that, when a card is marked as complete, it is archived or moved to a list that is archived, as well as becoming invisible on the board.
But the ask here is for start date. I too think that adding the option of a start date at the task level is a highly required and hopefully anticipated feature.
I will be looking for other options for my team if this is not available.
