I have seen the prior question - but it didn't focus on what I would see as the value to each site, and by inference, the wider SE network:
As a representative of a site, possibly connecting with professional organisations and potential sponsors there would be value in having a consistent front, so that official communications could come from eg. a security.stackexchange email address.
It would also give some moderators the ability to entirely separate their work communications from their volunteer SE communications - this could be a good thing in some environments.