In Germany, the broker fee is a one time payment. That makes for an complicated calculation, as rentals are usually not fixed-term, and people will expect to stay for several years or decades. Is 900 EUR per month + 1,800 Euro one time payment better or worse than 1,000 Euro per month without a broker fee? Most people do not think like that. 900 < 1,000 Euro, end of story.

Of course, the real problem was that the landlords did not both to negotiate the brokers fee. There was a maximum broker fee defined by law, and everyone just charged the maximum. Not anymore!

The broker fee IME is usually an extra month of rent due at the time of signing the lease. Because in certain cities the vast majority of apartments will follow this convention, and it's not really something that will be mentioned until you're in the process of formally applying (unless you ask), I don't see it having a market effect in the same way that raising the rent would.

Landlord already charges as much as they can. Increasing his cost will not magically mean that they can suddenly charge more than they can.

German landlords can't set the rent as high as they want, for starters. At least not in Berlin with the Mietpreisbremse capping how high rent can be relative to the past, and to the neighbourhood.