How do brokers negotiate the purchase price?

But it's just common that he says that - when he sees that there is something to gain?

Smells like a conspiracy theory
Or black money client :P

Well, yeah, I could have guessed that. I thought there might be some kind of customary rule that the broker usually stays involved.

Honestly: the broker doesn't want to work unnecessarily, he wants to make money quickly without much work.
In commission activities, the bird in the hand is usually worth more than the pigeon on the roof.

I have never had to tell/show the broker how much equity I have.

You just have to make it clear by being committed enough that you are capable of paying the purchase price.

A confirmation from the bank, which is usually then necessary, also says nothing about the equity.

Orient yourself to your budget and what the property is worth to you. If someone pays more and gets the contract, keep looking. Creditworthiness only helps you if the competition is low and the seller wants the money quickly and without complications. "XXXT€ is our offer, we can provide the money tomorrow in an escrow account. You just need to agree now."

If someone comes and says, prove to me that you can pay for the house, I say, no, that's none of your business what my bank account looks like. Just assume that I can and will pay. And if he insists further, then always again just assume.... you definitely have to assert yourself and show your dignity, yes, also maintain a certain distance. Psychology also applies in business. And seriously, the real estate agent really has no right to know what my bank account looks like. And in my entire life I have never provided such a "proof of ability to pay." Always said, I don't do that, period. I will pay, if you don't trust me, then we are both in the wrong here. Karsten

this bidding process and the "stinginess is cool" mentality disgust me. everyone thinks they can hold private auctions with their property. i hope some people end up getting their comeuppance because of this.

Even in the regular purchase process, the broker can whisper to the seller that there might be more to get from the buyer. I see no difference from the bidding process.