The veteran Scottish manager did not have his contract renewed at the end of the season.
Instead, the club spoke with various coaches while Moyes was in the job and eventually hired Julen Lopetegui.
I think it is incredibly tough that everything is so public now, everyone's got a smartphone that people can take pictures on," Sullivan told The Sports Agents podcast.
I think everyone's hope is to do it as dignified as possible, nobody wants to embarrass someone, everyone wants the best result for everyone.
And to be honest with you David Moyes left at the end of his contract, it wasn't like we terminated the contract early and I think when you look at Dad's (David) and Karren's track record is that they always keep managers until they have to part ways."