House manufacturer threatens to stop work

So once again... for explanation

there were already discrepancies in the planning and we had several emails and escalation meetings

I have to mention here that writing an email when you want to complain or express dissatisfaction is certainly easier than shouting, but an email is the worst form of communication overall.

I write such an email after every conversation,

I did not mean correspondence, recording of conversations, agreements, or the like, but what I also wrote:
when it comes to "complaint or dissatisfaction"! I can also write deficiency or defects. That might be clearer, because you don't do that by email, as the form of an official notice of defects is not then observed. Because that sets a business process in motion.
An email could be waited out. But yes: relevant for the later lawyer-trouble file. But at the latest then, casualness will be put on the table.
It may be that emails have gained a new significance in the last 10 years and are sufficient even for smaller transactions, but when it comes to termination of expensive contracts or a notice of defect in house construction, I would question the form of an email – furthermore, a letter has completely different, i.e. formal components.

It may be that the email has gained a new significance in the last 10 years

That's how it is. When a dispute between the client and the construction company ends up in court, the course and escalation of the dispute are analyzed. Emails and behavior play a decisive role in a judicial assessment. Email communication now plays an important role as a written medium. My advice is to conduct this from the beginning properly—nothing more. Incidentally, in the business world, even larger transactions are settled sufficiently legally securely via electronic media. Paper is still used, as you write, for contracts, deadlines, and terminations—but hardly anymore for minutes of meetings—even when a lot is at stake. Most of the paperwork that a private person has to deal with primarily serves to protect a non-private counterpart, who has to comply with extensive consumer protection conditions in this relationship.

Email communication now plays a significant role as a written medium. My advice is to handle it properly from the start - nothing more.

And what do you think? Should an official complaint about defects, which is otherwise sent by registered mail, be sent by email nowadays?

Yes! Dr. Max Greger on this: For contracts between entrepreneurs and consumers – since October 1, 2016, according to § 309 No. 13 of the Building Code: Standard terms and conditions may no longer require written form for declarations, but only “text form” (§ 126b Building Code) (this includes, among others: email, fax, scan)! With this, the legislator has legally regulated what had already been decided by the courts.

Should one nowadays send an official complaint about defects, which is otherwise sent by registered mail, by email?

As written: Email is useful to create clarity and a file record - if in doubt, even without a response from the recipient. That was the starting point. I have nothing against postal mail. I am able to think "both - and".

In contracts between entrepreneurs and consumers – since October 1, 2016, according to § 309 No. 13 Building Code: GTCs may no longer require written form for statements, but only "text form" (§ 126b Building Code) (this includes, among others: email, fax, scan)

Thanks for looking that up.

Oh dear, terrible story...

What is your main goal now? What do you want to achieve?
- Complete the construction as quickly as possible
- Fully eliminate all defects (balcony)
- Do not pay any additional cost claims
- End the contract as quickly as possible and continue with your own craftsmen

Of course, all points would be desirable, but on the other hand, it should not drag on forever either.
How much money is involved at the moment? How high are the additional claims? How much money has been withheld due to the defects? How much is still outstanding?

What approach does your lawyer recommend?
Personally, I would already send the next correspondence through a lawyer. Otherwise, I believe you won’t get anywhere with the [GU] at all. He is also unfairly blackmailing you: You have to reserve rooms for craftsmen (!) and then he threatens not to come if you don’t pay extra costs... totally crazy. Anyway... no lawyer without a clear strategy and don’t demand everything at once.