House Purchase - Complicated Situation

So you would receive a recognizable advantage, which in itself is not a problem. But – what advantage would the father have?

The father's wish is that his house remains in the family and is not sold. Even though I myself don't have such an attachment to objects, I can understand that. He may have built it himself and wishes that his future grandchildren play in the garden. Clearly, it is not a guarantee if he transfers it now to one son, but if the others already have their own houses (presumably new), it is more likely to remain in the family with a transfer now than if after the father's death no one needs a house anymore but everyone wants the maximum cash. I certainly see the possibility to fulfill the father's wish here, even if he does not have a financial advantage from it.

consider, namely what the father himself wants

But this is nowhere to be read. It is simply about what the OP imagines. If it were the father's wish, there would be no problem for the OP at all, meaning it would not be his issue.

The father's wish is that his house stays in the family and is not sold. Even though I personally don't have such an attachment to objects, I can understand that. He may have built it himself and wishes that his future grandchildren play in the garden. Of course, it's not a guarantee if he transfers it now to one son, but if the others already have their own houses (presumably newer), it is more likely with a current transfer that it will stay in the family than if after the father's death no one needs a house anymore, but all want the maximum money. I definitely see the possibility here to fulfill the father's will, even if he doesn't gain a financial advantage from it.

Sure, I understand that and apparently that is also ONE wish of the father, which in turn can be seen as an advantage for him. What I have difficulty with, however, is the search for a 1-room apartment or 2-room for the father. I have experienced myself that parents hold back excessively and basically "sacrifice" themselves for the children, but of course they also have wishes and fears. So far, I am missing that information. If the father then lives somewhat further away and special care is not mutually agreed upon, he will be sitting alone in his apartment... therefore, for me such details would be important components of a possible agreement. Money would be rather secondary for me as a father.
I would grant the young people who take care of me/us at any time a significant advantage; the others would or would have to understand that, since my (future testator’s) well-being is my main priority.

and wishes that his future grandchildren play in the garden.

I absolutely understand and agree, grandparents are usually happy to be there and experience that. But... do the children (heirs) want such permanent proximity to the father and possibly any obligation that comes with it for him?

I definitely see the possibility here to fulfill the father's will, even if he doesn't gain a financial advantage from it.

Absolutely – only, I am also sure that this is only ONE will/wish of the father and not the only one; it is unlikely that he wouldn't also want closeness to the family and security in old age.
As already said, the younger generation tends to focus more on finances; an older person probably looks less at money and more at avoiding loneliness and being close to the family.

I will leave all other aspects aside; the topic is, regarding the people to be considered and, among other things, tax consequences, the typical case of six people bringing seven opinions into play, and mine would only make the stew unnecessarily more colorful. But I do want to highlight one point, or rather, I consider it worth exploring more than has been done so far:

The point is that my father wants the house to stay in the family.

Where does this wish come from, and is it realistic? – what happens to your father if you remodel the house differently than he imagined?
I have not yet found out here who this father actually is and why only the father is mentioned: has the mother already died, or is the father divorced / abandoned by the mother?
My fear is that the house could be symbolically overloaded, i.e., have a museum function – as a place where the family was still intact or the marriage was still intact. And then you come and throw the kitchen onto the recycling yard, where mom cooked dad’s favorite dish – then he might as well have handed the house over to strangers!?
Or was the father a bank robber and the house has to stay in the family because the loot is cast into the garage foundation?

Once again, many thanks for all the numerous contributions and opinions; that really helps me a lot to consider this challenging situation from all possible angles.

I now think, as confirmed by some contributions, that the mutual rental scenario only really benefits the state. Whether I would actually have a tax advantage from it is questionable.

The cleanest solution, I think, would be this proposal:

You write that you can manage financing up to 550k. Let's assume the house is worth 600k, then one possibility would be that the father sells it to you for 400k and at the same time a will is created in which the remaining 200k is recorded as a "gift" to you as an advance inheritance with waiver of compulsory portion (i.e. you inherit nothing later!).
From the 400k the father receives from you, he buys himself a nice new apartment. This can then be inherited later by the other two brothers.

As a first step, it would of course make sense to have the house appraised by an expert, whose value can then be used to plan the further steps.

I can well imagine continuing down this path.
Many thanks to everyone.

The first step would of course be to have the house appraised by an expert, so that you can plan the further steps based on that value.

Overall, I would approach it that way as well. It might also be an option to agree on a market value with the sibling(s). The appraiser also costs a four-figure amount and ultimately he is just "appraising" as well.