Literally being in the tabloids is a minor concern, yes, but hopefully you are able to generalize beyond that.
They aren't "silly" requests, and the realtor has a legal obligation to present offers to clients, so I'm not sure how you want your hypothetical screening to work.
Indeed, I recently was asked to sign an NDA to view an off market home; the seller was (apparently) a local business owner and didn’t want folks thinking the business might close — in many industries confidence in long term staying power is critical.
They aren't "silly" requests, and the realtor has a legal obligation to present offers to clients, so I'm not sure how you want your hypothetical screening to work.