Yes, Diana could only ever have been a Queen consort like the current Queen's mother was when King George VI was on the throne; she could never have been our actual Head of State.
Putting that aside: why would the establishment despise the idea of Mohamed Al-Fayed's son fathering a half-brother of Princes William and Harry?
The Queen and Charles, for all their faults, do not strike me as racist or Islamophobic. Indeed, Charles regularly visits Mosques as part of his interfaith work and the Queen recently had a Muslim prayer room built at Buckingham Palace.