You do not have €6,000 but €4,750 and want a maximum rate of €1,300. However, you have to adjust to €2,000 including all additional costs. Period.
I am usually very conservative about such things, but in this case here.... 6,000 net, the house is not excessively expensive... the income extremely secure, with a family also mostly easy to estimate that they want to stay in the region..
To hell with it, you need the house now with the children, what are they supposed to buy in 5-10 years? Then it is (exaggeratedly put) almost again time for an age-appropriate bungalow.
It is reasonably affordable and you want to have the quality of living through your own house NOW, and not in 10 years. You can also let life just pass by. Financially, it would probably be better to save equity first, but it is not always just about optimizing the accumulation of capital in life... The increase in quality of life as a family with 2 children through your own house + garden (when it does not mean you have to accept severe financial cuts) is enormous, especially when the children are still young!
10 years fixed interest period? But after that there is still a huge amount left called residual debt! Caution, caution, you certainly don't want to inherit any debts.
On the other hand, after 10 years an even larger amount is outstanding if you have a long fixed interest period and thus a correspondingly higher interest rate. A difference of 10,000 euros in the remaining debt adds up quickly. The original poster can probably tolerate the risk well and service the loan even at higher interest rates.
By the way, I wouldn't be so sure that all the financings with 20 or even 30 years really last until the end of the term. Who knows which president will decree by then that these financings are immoral to save the banks from the citizens. Thirty years ago the wall still stood in Germany, that is a long time.