But it sounds like you were dealing with an individual landlord, that's kinda rare. They might be more willing to use common sense and not ask useless documents from someone who just arrived in the country.
On the other hand, big property management companies will typically not bother with an application that doesn't tick all the boxes.
On the other hand, most landlords and property management companies absolutely hate applications from people who they know are going to leave after a fixed term contract.
Further, as of GDPR, you are entitled to all information from the agencies for free. Prior, you could obtain it for a small fee (£2 by post if my memory serves me correctly).
As far as I'm concerned, SCHUFA is basically a scam I am compelled to fall for whenever I want to rent somewhere.