Because the basic purpose of that kind of identity politics is to gain status by distinguishing yourself from others. There’s no point in asserting a British-American identity, or a plain American identity, because nearly everyone can do that. So you assert something else instead. Anyway Biden does seem to be fairly heavily Irish-by-descent tbf, it’s way less tenuous for him than for other presidents.
I’ve found pretty much the same here, among certain right-thinkers. My neighbours did a DNA analysis for their kid for his 16th and the lad was a little bit gutted when it came back ‘100% British’. I told the story once for amusement when I was out with some liberal types and it was like dropping a grenade into the conversation, the table was falling over itself either to stress its own foreign DNA credentials, or else to even deny that being 100% British is possible. There’s no exotic points in being British.
According to the Natural History Museum, the original Brits are Homo heidelbergensis (about half a million years)
They are all descended from Homo Antecessor who likely migrated here via Spain according to tracks found aging about 750-900,00 years back.
Then Neanderthals, followed by Homo Sapiens.
I think the Neanderthals still remain today in small enclaves such as Grays, Canvey Island and Hartlepool.
(probably unfair on Neanderthals if I'm honest)