Honestly, they’re all good. But for the future:
1) Kings Arms basically serves an authentic menu for the wealthier clients (not for today, I mean it’s an “upper crust” colonial menu that the wealthier people would have eaten at the time),
2) Shields is actually less authentic as it includes things like hamburger (so whoever told your mom that was full of BS or didn’t know what they were talking about), but it has some authentic lower class colonial food that the poorer people would have eaten,
3) Cristiana Campbell’s is basically all seafood and is a mix of low and upper class colonial seafood dishes as well as modern stuff like oysters Rockefeller which doesn’t fit in with the colonial setting
4) Chownings is basically all simple pub fare which includes traditional British food like bangers and mash and bubble and squeak with colonial fare like turkey trenchers and corn chowder and a couple of modern pub fare items like portobello sandwiches. But it’s really more of a traditional British Pub with some nods to early American cuisine, so very much what the pre-Revolutionary War would have eaten.
So like I said, I prefer Kings Arm and Chownings, but all four are perfectly fine. And if they all have the same menu I doubt there would be much difference between the four in terms of quality.