The contractually agreed construction manager will not be provided by the construction company.

I also think that the construction company should have assessed this risk themselves. Now, it is not impossible to employ this construction manager, just more cumbersome and/or expensive but not impossible for them. However, that cannot be the problem of the clients. A clear case of "being right - getting the right". Ultimately, you cannot force the construction manager onto the construction site; maybe he himself does not want to do that anymore because of his move. Canceling the entire house construction contract will probably not be an option either, and a "forced" construction manager might not enjoy it much... Negotiating a sum will probably be difficult; maybe they can understand you in a direct conversation and throw some small obstacle in your way elsewhere. What does the company itself say about this apparently non-compliant contract clause or how do they want to deal with this partial non-compliance?

Now we have received the order confirmation from the construction company and a day later the call that it is not possible to get the desired site manager, as he now has his main residence in Bavaria and would therefore have to drive 5 hours to us. However, there are 2 alternatives in our area.

Which two alternatives: two good construction-accompanying experts or two other general contractors?

The site manager is an employee of the company. I would not care about him in case of doubt because he is just a subordinate anyway.

Even a highly praised contractor "site manager" cannot replace a site manager without quotation marks. I would always hire my own site manager and largely disregard the "site manager".

Even if one can question the sense and purpose of fixating on this one man. As long as the site manager is employed by the construction company or at least receives a large portion of his contracts from there, you can of course "force" him.
The employer can order this and the employee has to comply.
Sleeping one or two nights a month in a hotel instead of at home is probably reasonable.

As already mentioned, the higher costs for the company do not play a role here.

A completely different matter is the interpersonal level. And on this level, it makes no sense to have an annoyed site manager who, due to distance, can only come by every few weeks and already before construction begins picks fights with the construction company about trivial matters.

Especially since your acquaintances are probably laypeople and did not have their own expert? And even if...
Then you don’t even know whether the site manager is really good, or if your acquaintances just got lucky with the craftsmen.

Sleeping one or two nights a month in a hotel instead of at home is probably reasonable.

Good joke. Additionally, just on a brief visit, even a so-called site manager is an even more negligible figure.

Especially since your acquaintances are probably laymen and didn't have their own expert? And even if...
Then you don't really know if the site manager is actually good or if your acquaintances were just lucky with the craftsmen.

That's right. Freely after Mel Brooks in Spaceballs: "Forget the so-called site manager..."!

How to deal with this breach of contract (if it even is one) or what is reasonable for you or the general contractor, I will stay out of that. Has the contract signature already been some time ago so that the gentleman has since been transferred?

I find the prioritization quite odd, however.

An important aspect was to get the same site manager as our acquaintances, from whom we have the recommendation, as he was praised to the highest degree.

One should look at a company’s seriousness and performance, then of course at trust, which builds and sets through further factors. Then healthy financials and clean construction sites should not be underestimated. Employees, including site managers, are often replaced. Usually, everyone is replaceable. Then sympathy comes into play, that one gets along better with one person and another with someone else. For me, the site manager was relatively unimportant: the logistics were handled from higher up, others built the house, i.e., the craftsmen. I’d say: our site manager was different from us, had no sense of humor and… whatever: we nevertheless built a beautiful house. At that time, another company, i.e., Heinz von Heiden, had very big problems retaining its site managers. That might be due to the Heinz von Heiden philosophy. Our neighbors complained, not everything went smoothly. But they are now living in beautiful houses despite multiple site manager changes.

Whether one should attach so much importance to friends’ enthusiasm that one values it more than other reasons for building with the company, I do not see now. And as already says: ultimately, the company pays the site manager’s salary, not you.
I really have to think now what our site manager would have had to do to be praised by us “to the highest degree”!?! For us, it was more the craftsmen who implemented creative details and wishes that were not ordered.
If I were you, I would reconsider your dissatisfaction and demand, whether it is justified like that.

How to deal with this breach of contract (if it even is one) or what is reasonable for you or the GU, I will not get involved in.

A lawyer – whom I would consider pointless to consult here – would probably talk about main obligation, secondary obligation, proportionality, and possibly objective impossibility and point out that the house could already have been occupied for two years before a revocation or special termination right derived from the disappointment would be clarified in a third instance. I understand this disappointment, but I also see an emotionality in it that has no place in an investment. The so-called construction manager can harm no more than help. The fact that he is "included" in the price doesn't make much difference. Find a construction supervisor who is recommended in any case and, if he is good, also worth the money. For the standard contract aspect, the contractor’s site manager is sufficient, and for all other tasks, he is irrelevant if you involve a construction supervisor.