> Leaving aside the fact that rental markets already have a highly asymmetric power dynamic
> They'd probably decline to provide them and refuse to rent to you, in the same way as you can decline to provide them and refuse to rent from them.
This sounds like the definition of an asymmetric power difference to me. Sure, I can refuse to provide financial checks or a guarantor, but that only hurts me. Asking for that information from my landlord also only hurts me as a landlord is likely to avoid me (and tell the other landlords he knows) and go with someone more compliant.