That the answer. Pay people more to do the jobs nobody wants!
At this point I’m not even sure if big companies read cover letters at all.
I don't see any real reason why this is still required, or why it matter that the sex of a person be present on an ID document. The purpose of the ID is to identify the bearer. Most ID are now biometric, meaning the identification relies on unchanged physical features, like geometric relationships of the face.
A simple photo should be enough to confirm the ID of the bearer.
> If you want to make progress in any area, you need to be willing to give up your best ideas from time to time. [...] Medawar notes that he twice spent two whole years trying to corroborate groundless hypotheses.
Unfortunately, being willing to scrap your idea is only one part of the equation. Securing funding after 2 "failed" post-docs is an entirely different matter.