Laura It would in the US
Hmm, I would consider it a big invasion of my privacy if just anyone could check the health of my finances at any time.
But I certainly believe that privacy legislation is stronger in (western?) Europe than in the US.
If the Proof of Income only shows that you have a high-enough regular income, and not exactly how much, that would feel a bit different for me, also because if it is asked, I can still decide if I want to share that info, and with whom. That would seem more balanced between my privacy and their need for financial security.