Initial amount of cash needed to buy a house.

Well, I still doubt that they will be completely abolished. Here in Düsseldorf and the surrounding area, it is always 3.57 or 4.76%. For reasonably sought-after properties, it is actually always 4.76. (I deliberately do not lower it because of the reduced VAT. Most brokers here don’t do that. They pocket that extra.) In practice, the buyer usually pays it alone.
I think no seller is keen to put 4.76% on the table. Even if it "only" comes off their achieved sales price.
From January, real competition will finally arrive. That also means that besides qualitative features, the price will ultimately be decisive. And that benefits the buyer equally.

Nevertheless, THIS CAN mean that the seller’s share will be included in the price. Then the question is at what percentage the commission will settle.
If it is less than half of the current commission, it is good for financing. Anything over half is rather bad.
Example (all rounded figures):
Today:
Commission (to be paid solely by the buyer): 4.76% = 14,000 euros
House price: 300,000 euros.
Bank valuation: 280,000 euros

From 2021:
Commission (each by buyer and seller): 3.57% = 11,000.
House price: 311,000 euros
Bank valuation: 280,000 euros

So you have saved 4,000 euros of equity. But the gap between the bank valuation and the price has increased by 11,000 euros.
And guess where the bank usually wants that lie covered from...

Briefly the example if it would settle at 2%:
Saved equity: 8,000 euros
Gap between bank valuation and purchase price only increased by 6,000.

The valuation of the house by the bank will actually adjust upwards. A flat safety deduction of ten or 15% of the purchase price equals the lending value.

What kind of profile is that? None at all!

The bank's valuation of the house will actually adjust upwards. A flat safety discount of ten or 15% on the purchase price equals the loan value.

Well, if it's that simple. Why do banks even commission appraisers if they can just take the purchase price as the basis for assessment? And then the appraiser will simply include the real estate agent's commission in the house value calculation?

Well, if it's that simple. Why do banks even commission appraisers when they can simply use the purchase price as the basis for assessment?

Because they have to and because there are regulations on how to determine a lending value.

And the appraiser will simply factor the broker's commission into the house value?

The appraiser/the bank only see the purchase price and do not see a separately factored-in broker's commission. And whether a property costs €10,000 more or less is irrelevant for the valuation, since there is no single value of the property.
I see so many appraisals (for large properties) where you can place the seller’s appraisal, the buyer’s appraisal, and the bank’s appraisal side by side – the seller’s appraisal says €50 million and is correctly calculated, the buyer’s appraisal says €40 million and is correctly calculated, and our own appraisal says €44.5 million with correct calculation. It always depends on the perspective on the property and is to a large extent subjective, shaped by the personal opinion of the appraiser.