Like with any company or industry, the answer is pretty simple: “It depends”.
Any time you have somebody whose duties include being in the public eye, the company that employs them are especially keen on them maintaining an image that is acceptable to the public. As a food critic who does not publish anonymously/pseudononymously, if unflattering information comes out that hinders his ability to connect with the public in his position at the company, then it’s simply good business to terminate his employment.
If he were some random line cook or back office type grunt, I’d say no. In his position as “person who writes things for public consumption”, I’d say yes. In this particular case, can him and let’s move on.