Analysis of our financial situation

When builders take out an affordable loan with about €4000, after a repayment of €1500, €2500 remain for living expenses, which is usually sufficient.
In your case, with about €7300, you could invest around €4500 in your loan.
Whether you want to do this or whether your household budget, see personal additional costs, allows it, you have to calculate yourself. There is too little information, see
https://www.hausbau-forum.de/threads/welche-Infos-um-Finanzierung-Angebot-bewerten-zu-koennen.18200/

When I take on a project like this, it has to be a bargain. I don't know your area, it doesn't read like it. You will get the financing. With children, the belt just has to be tightened more.

Well, he lives in BW, we in Bavaria.... here in our area, for 560k you unfortunately only get something where you still have to put in 250k ... especially if the plot of land is a bit bigger

I also believe that the financing is feasible - the question that arises for me is: is it sensible? A building from the 50s - with all the disadvantages of construction methods at that time (which relates to room layout, sanitation, electrical as well as simply the building fabric) - I do wonder whether tearing it down and rebuilding wouldn’t be more sensible. And then the price would be too high. Is the house so cute and beautiful that you absolutely want to preserve it or is there even [a historic preservation order] on it?

Well, he lives in BW, we live in Bavaria .... here in our area, for 560k you unfortunately only get something where you still have to put in 250k ... especially if the plot is a bit bigger

That would be a sign to me that it is the wrong time. Where does this compulsion always come from to knowingly pay far too much? Because everyone is building? Financially it's possible, but the question of sense is important. It also feels different now living with very generous surpluses (that also does something mentally) and then afterwards tying up large parts in the financing. Real estate makes you immobile anyway, but when one goes part-time the other only then understands what it means to carry a financial burden. Even if it is objectively okay. You always have to see where you are coming from. You from 4k€ surplus down to <1000€ surplus, which mainly one alone will have to bear. That is quite different.

I don't really see a problem there - especially because of the solid equity.

With a 650k loan at 5% annuity, that would be about ~2700€/month. So what?

It just has to be clear that the wife MUST go back to work fairly quickly, and in addition, I would probably try to build up a buffer until the birth of a child in order to better bridge the time with one income.

What would be clear to me in this scenario, however, would be an interest rate lock of at least 15-20 years. That would be too risky for me with interest rate changes and the amount.