Is a single-family house in the Stuttgart metropolitan area still affordable?

So either you come from a different culture where it really is like that, or nobody tells you what they received from their parents, especially because they know your opinion about it.
Apart from my best friend, no one knows the amount I got from my parents.

I wish my parents all the best and don’t expect anything in return.
What seems impossible for you to understand is that some people have enough money to both live nicely themselves and also give something to their children.
And what you can understand even less is that different things are important to different people. My parents have never been to a restaurant except when they were forced to go to birthday parties. Just like me, they also prefer eating at home. My parents don’t want to travel anymore because they feel more comfortable at home and find traveling too exhausting. No one in my family has ever cared much about clothes. I get super annoyed when something breaks and just replace it with the next best thing I find on a common website. I find the cinema boring because I find it cozier at home.
And I could give you many more examples like that.
What I do find really great and that gives me a lot of joy is spending time with my children and also making them happy. Giving also makes one happy.
Likewise, my parents are much happier that we built a house than about all the other things they could do with the money.
By the way, they now also donate a good part of their income to charitable causes because that gives them more than yet another unnecessary purchase for themselves.
And I find that sensible too because, unlike many others in the world, we really have enough...

And where we also disagree: A house is NOT a luxury good for us. My parents have a beautifully dream house on a good plot. Owning a house is part of life for us. I can think of 1000 other things I’d rather give up.
But everyone has to know for themselves what is important to them.

It would be interesting to know what you would really think if your parents had 1 million lying around in their bank account. Would you seriously find it sensible for them to completely spend the million?

I would disagree with you to the extent that as a career starter at 29, one might still be able to make salary jumps. Of course, you can’t firmly plan on that. I still stick to all the statements, "you can’t live on so little." Let’s calculate 1800+3700 (presumably the man’s net income is higher than before, as the total income decreased). So 5500€. After deducting a 3000€ installment, you have 2500€ left. Why can so many people here not live on 2500€/month, even deducting the cold rent, for just 2-3 years of parental leave? How does that add up when the average German household has 3300 net and still has to pay cold rent? Where is my thinking error?

I also find it a huge difference whether I ask my parents for money, basically mooching directly, or whether they say on their own "we would like to support you." Then it was their free choice, and I don’t put them in any kind of pressure situation (what do we answer now?).

Yes, my parents also supported me when buying the house. In the end, they covered the ancillary purchase costs. But: it is explicitly not a gift, it is a loan. Interest-free, yes, but they want it back.
And they are getting it. The first sums have already flowed back, one year after moving into the house. Of course, I will have to pay on it for a while, no question.

My parents are retired, doing okay so far. They use their savings, as long as they are fit, for travel and in case a new car is due sometime. They offered to support me on their own initiative. That’s what made the project possible in the first place. But of course, I make sure to give back so much that they lack nothing they would otherwise have afforded.
Still, I stand on my own two feet.

Your error in thinking is the following: with income, usually the demands also increase. The best example is the available income as a trainee or student. Most get by with 800 euros net. After finishing training or studies, more is earned and demands rise for the first time and more money is spent on consumption and partly also on insurance and retirement provision. Often a consumption expense also leads to further costs, e.g., during training a Golf 2 with liability insurance, afterward a 7 Series Golf annual car with comprehensive insurance and regular inspections.

On the topic of accepting money from parents for property purchase:

My wife and I consciously refrained from asking our parents if they would financially support us when buying the apartment. We find it mega embarrassing to still have to ask your parents for money at 30 years old in order to afford something. That reminds me too much of school times...

And yes, both sides have enough money and it would not have financially hurt my parents-in-law if they had given us the entire money for the apartment as a gift. I wish my parents and parents-in-law a long and healthy life and they should use their self-built and earned assets for themselves and are welcome to completely spend and live it up until their death, since it belongs to them and therefore we have zero claim to it.

Yes, I apparently come from a different culture. One where, as an adult, you no longer take money from your parents, but instead treat them to a vacation or give them fully automatic coffee machines for their birthday. After all, they have raised the children for at least 18 years and had costs higher than a house. That is why I/we as my family are responsible for ourselves; especially financially. Of course, there are exceptions (illness, etc.). Luxury goods definitely do not count among them. My daughter would never come up with the idea of asking me for money for a house. Either you can afford it or you cannot. I would also like to drive a Mercedes GLS, but I can only afford a GLC. But I don’t run to my parents because of that.

And yes, that is why I cannot understand it. I grew up in a giving culture when it comes to adult children. But apparently that is no longer the case...

I grant my parents everything they want and don’t expect anything at all. What apparently doesn’t make sense to you is that some people have enough money to both live nicely themselves and give something to their children.