The planner does not perform any calculations according to DIN 276

Of course, I do not want to pay the 6,000 euros demanded by her from the final invoice initially (total amount 67,000 euros),

Then withhold the money and justify it with the incomplete performance within the objection period. You write "initially" – what else do you want?

I have to admit, little by little I somehow have a completely different picture in mind than what the OP is trying to describe. Now 80 k€ suddenly become 67 k€. How much of the rest is then correct? Was there really only one offer per trade everywhere? Is it really 12 k€ more for the scaffolding or is that the total amount? Is there really no cost breakdown? There also seems to have been some incident, since the architect no longer dares to go to her clients without an escort.

What if she is not an architect at all-

You can easily find that out with a call to the Chamber of Architects?!

: What do you really want to achieve? It’s hardly about a retroactively filled estimate.

I’m wondering that too. In the end, the renovation is finished, right? If invoices still come in from her for services she did not provide, you can get a lawyer to enforce “his right,” meaning your father’s. That means: your father’s duty. your father’s mandate. It seems to me you want to fill your father’s account with “rights” after the fact, although it would have been necessary to act early on back then? Much of it simply has to be written off as a learning experience afterwards – it doesn’t matter whether you naively paid invoices or because it was all getting out of hand.

Hmm well, you don't have to put up with everything.

If services were not provided or were even provided incorrectly, there is only one way:
- Withhold payment
- Remind about outstanding services
- Remind about errors, if any, and attach a price tag
- Reach an agreement

If you can't do that alone, you go to a lawyer who will gladly help for a corresponding fee.

My father really paid every invoice within minutes - and also wants to reach an agreement with the roofer, however, this is not my father’s fault, since the planner scheduled all the appointments with the craftsmen. Why does he now have to pay €12,000 for scaffolding that was not used - not even by the roofer who set it up?!
Honestly, I am sure that we are even in the area of compensation for damages here - so it is difficult for me to pay her anything more now, especially since no cost determination has been made yet / and she simply claims "just a bit expensive? But it looks nice" [...] What if she isn’t even an architect - but only a “specialist planner”? Is one allowed to enter into such a contract covering all service phases - according to HOAI including annexes?
Is anyone allowed to do that?

Only those who have demonstrably completed the relevant studies are allowed to call themselves architects or civil engineers; without this, they are not allowed to work freelance in this profession either, and they automatically become members of the respective chamber (of their federal state of residence). This professional association is not an open club for everyone; only holders of the relevant professional licenses can (and must) be members. Architects also include their chamber membership number on every contract and every invoice, as well as on every building application or certification to funding providers. No one can simply call themselves “architect” if the nail salon didn’t work out. Using HOAI with the motto “great, then I don’t have to invent my own GTCs and price list” would be possible but useless if one lacks the authorization to practice as an architect or engineer. “Poaching” as an architect would be an administrative offense and, in my opinion, would also invalidate any corresponding service contract (I already mentioned the nasty word).

If the architect was authorized to commission the roofer and the roofer provided the scaffolding beyond his own needs, he is entitled to payment accordingly under the contract. If the father is not responsible for the extended duration of the scaffolding provision and the architect acted in bad faith here and harmed him, the roofer should not have to bear that. If your descriptions – woe to anyone who misleads their lawyer – fully correspond to the truth, then compensation for damages here seems to me to clearly outweigh the remaining fee at least. Now it is Wednesday and lunchtime; the lawyers are on the golf course. So call there directly tomorrow. I’m about to leave here myself now.

Hmm well, you don't have to put up with everything.

is that now the answer to mine? I wrote it:

you can get a lawyer to enforce "your right," namely that of your father.