Can an average family afford a single-family house at all?

As Jochen has already correctly indicated, there is no ONE nationwide valid price. Building is very different regionally; even differing again within the individual federal states. I would also not want to claim that building a house in Saarland is significantly cheaper than in Bavaria; assuming the same equipment and execution.

Prices have - as I would assess it - increased by a good 15% in the last 3 years. When I look alone at what I have to pay today for copper and even multilayer composite pipes (used today in sanitary pipe installation instead of copper), it makes me sick. Even a small shower tray of 1.00 x 1.20 m currently costs €730.00 net in equity. This runs through all trades.

More or less yes.

Rhenish greetings

Ok, 15% is of course quite a bit, although as I said, the comparison would be KfW70 old vs. Energy Saving Ordinance new. And we also no longer have copper pipes, just as tiles for €50 per m² (cheapest online price) are not necessarily standard, I would say.
Why a barrier-free shower with a shower channel is so much more expensive than the already expensive shower tray I don’t understand anyway.
An Aco shower channel costs about €600, embedding it in screed takes maybe an hour of work plus sealing and tiling a maximum of another €200.
Partition and shower head remain the same...

It would really be interesting to see what an estimate would look like today, whether that would be compensated by the even lower interest rates...

whereas, as said, the comparison would be KfW70 old vs. new Energy Saving Ordinance

I don't see it that way. The comparison must be from Energy Saving Ordinance to Energy Saving Ordinance house. If I didn't want to build an efficiency house last year, I can't use an efficiency house to compare prices the following year. That's comparing apples and oranges. Whoever wanted to build a KFW70 house last year would have signed for it and submitted everything for the funding. Whoever didn't want that must now basically build a KFW70 house but receives no funding.

It's not about subsidies or financing, but about the pure house price, meaning the necessary technology, insulation, etc... And for example, 60 kWh/a = 60 kWh/a, no matter how the house is classified according to which energy saving regulation.

Okay, I'll phrase it differently: I want to build a house according to the lowest energy standard that is allowed. Until last year, that was the Energy Saving Ordinance 14, now it's 16. To meet this lowest standard as of today, I have to make an additional investment (i.e., better insulation, etc.) Therefore, the comparison between KFW70 and Energy Saving Ordinance 2016 only makes limited sense; they probably differ by about 3-5%, meaning a normal price increase in construction.

That's why the price comparison between KfW70 old vs. new Energy Saving Ordinance, which would then be the technically similar house...

Only a few make this comparison, at least 90% simply want to build what they have to. At least that's the picture in our circle of acquaintances, we are more "exotics" with KfW55, and our general contractor had a similar opinion. This makes the house more expensive for this average.

Whether KfW55 is even worthwhile is questionable, if one might save maybe 20.-€ in energy costs per month compared to the "Standard" house. Since the Energy Saving Ordinance 2009, houses have been very efficient anyway...