And what costs [...] are we talking about here for a single-family house?
This is told to you by the Fee Structure for Architects and Engineers (HOAI), it depends on the object (more expensive, larger house is more expensive; also whether it is detached or has to be integrated into a building gap makes a difference).
Simply put: more expensive house, more expensive planning.
The most expensive architect is the one you "save"
(only the plan approved for permission without construction supervision)
As far as I know, only the architect owes you the "plan approved for permission."
Is a matter of negotiation for a single-family house,
Yes, the HOAI is a framework, negotiating is allowed.