This isn’t as stupid as it sounds. Anything can become an element of national identity. It can be ancient or invented 5 minutes ago. It doesn’t even have to be connected to people who were aware of your nation-state because it was created long after they died.
The problem isn’t that the BBC and the NHS are recent or that they are institutions instead of, say, natural phenomena. The American nation-state by necessity relies on achievements that are enormously more recent than those of European countries, yet it’s the first, and still the strongest, nation-state in existence. Newness is not in conflict with national identity.
The problem is that the narrative of the British national identity that ties it to the NHS and the BBC was embraced along class lines by people who despise their class inferiors. You can’t have a separate national identity for the fancy people. Trying to position your class identity (which is what “Britain is the NHS is”) as national identity only annoys the people to whom you are signaling that you are their class superior.