"I wonder if there's a space for business for an intermediary who would take on multiple builders for your single home, wait for them to do their price discovery dance, and then drop every one of them except for the cheapest one."

Having worked with subcontractors a race to the bottom is not a good plan. IMHO subcontractors cost is directly proportional to experience. Simple unseen things will get missed eg: wrong type of drywall in the bathroom.