I have a 1974 American reprint of Arthur C. Clarke’s 1957 short story collection Tales from the “White Hart.” On every reference, in title as well as text, the name of the pub is in quotation marks. The Wiki article lacks the quotation marks, though, as apparently did the cover of the first edition: Tales from the White Hart - Wikipedia
I’ve been in the UK several times and don’t remember ever seeing pub names in quotation marks when referred to in print. Was it an old custom?
I’m pretty sure a style guideline said that you used “” to indicate that the White Hart was a named entity, and not just a pale stag of no particular speciality.
That sounds like the most likely explanation, but it would be a very old-fashioned style guide if so: I’d associate it with Victorian habits. These days, surely the initial capitals that are there anyway are enough to indicate a proper noun.
Wild guessing: Sometimes pubs have names, but are often referred to by nicknames - C. S. Lewis often went to a pub named The Eagle and Child, but often called “The Bird and Baby.” Putting quotations around “The White Hart” might indicate that the “The White Hart” isn’t the pub’s real name, but just a nickname.
A lot of old pubs don’t even have official names. The place might have a sign out front with a picture on it, and everyone refers to it by a description of the picture. The same pub that one person calls the “White Hart”, another might call the “Snowy Stag”.