(Unfounded) fear of monthly burden?

As said, reduce expenses and also ease back on demands a bit, do they really have to be 500k? I don’t know exactly what prices are like for you, but we built 140sqm with just under 300k with upscale fittings, some own work, and a small plot of land.

Cost of living (everyday essentials): €1300

Well, hold my hat tight
I think this is where the problem lies.... €1300 per month for "everyday essentials." That means you spend over €40 every day... And I thought we were already living large with almost €600 in costs.
Do you have any idea where you’re spending that money? Do you only shop at organic supermarkets? Expensive cosmetics? Are medications included? Or do you simply not watch your expenses when shopping?

I don’t want to criticize you, but you need to address this expense category. Analyze it over the next few months.

The costs for clothing, kindergarten, and GEZ are included in the 1300 € for living expenses. I have calculated 800 € solely for groceries (or daily shopping).

Keep a household budget for the 1300 costs. There's no way around it. Expensive hobby, eating out twice a week?

Otherwise, it's complaining at a high level (sorry for the expression)
Equity generously available, civil servant for life + very good salary... plus some salary would probably come from your wife?!

Some couples barely have the income and almost no equity...

We have already saved something monthly, but then touched the savings again whenever sudden expenses came up ...

I calculate the expenses after the house construction (spread over the month):

Additional housing costs: €450
Insurances: €380 (including my private health insurance)
2 life insurances: €130
Living expenses (everyday necessities): €1300
Vacation: €200
Car (including fuel): €350
Miscellaneous (including commuting costs to work - are separate.., repair reserves, ADAC membership fee): €240

Total monthly amount: €3050. And nothing has been saved yet...

First of all, very good: You are already planning for additional housing costs. Of course, these depend on the size and equipment of the house. The bank calculates between €1.50 - €2.00 per sqm.

Regarding living expenses, ypg has already given you the ultimate tip: Keep a household budget book. Record every single expense in it - no matter if it’s a bread roll or an Amazon order. Even if it’s annoying (my wife constantly rolls her eyes). You should keep this for several months. Based on this, you can find out where the money goes - how much it actually is on average and where potential savings might be. Also, it makes a good impression if you can show the bank that you have your costs under control.

Without knowing how old you are, how long you have been saving, and whether your assets may have grown to €250,000 through special income (e.g. inheritance/gift), I dare to say that you actually save more than €10,800 over the course of the year. If I calculate €250,000 / €11,000 = about 23 years consistently saving €11,000 per year. Therefore - take a close look at your finances before you plunge into the building project.

OK 1,300 with [Kindergartenkosten]. Now for the tricky question. Does your wife earn nothing? What about the 4,000 €? Is that tax class 3?