And for a differing opinion......😉

I would neither remove any soil nor would I import any topsoil. What I see in the photos is soil that is devoid of much - if any - organic matter. And OM is critical in developing an active soil biology as well as providing the full range of nutrients plants need to thrive and be productive. And it will eventually improve soil structure and help with poor drainage. Instead, I would get my hands on as much compost or composted manures or any other OM as possible and topdress/mulch the planting area deeply. And I would do this at least one full growing season ahead of any planting to allow sufficient time for the OM to do its work.

It could well be that the marshy or wet conditions after a good rain is due to heavy compaction - since it was a parking lot - so loosening the soil or aerating/reducing compaction would also be a priority,

Contrary to poplar belief, asphalt is not a hindrance to successful plant growing, even with edibles. It occurs naturally (research bitumen), is heavily recycled and since petroleum based, is not take up by the plants. It generally has no adverse effect on plant growth (unless it's smothering them :-)) But like the previous comment, I see no obvious signs of asphalt present.