Private keys don't verify that an NFT was authorized by the owners of the painting. You're still relying on real-world trust and claims that the NFT you're buying was the "real" NFT.
If the owners of the Banksy came out a few days later and claimed that someone else sold a fake Banksy NFT, but they're now going to sell the real NFT, the private keys won't help.
I am not sure why I got down votes here. If you are buying a NFT for an artwork that is not confirmed to have been created by the author of that artwork then surely you are just stupid?
Only Banksy can mint an NFT for his artwork for it to be of ANY value. The same way that only Banksy can create Banksy art.
Public-private key encryption is how you verify that something was signed by the original author.
If the owners of the Banksy came out a few days later and claimed that someone else sold a fake Banksy NFT, but they're now going to sell the real NFT, the private keys won't help.