I don't think Harry and his apologists should be able to plead ignorance about his racism 'because of the times back then' or similar excuses, or say his (royal) family just talked like that around him either. Harry was born in 1984, at school in the 1990s and 2000s, and the Race Relations Act ('Section 1: Racial Discrimination') under Wilson/Callaghan received royal assent in 1976. It was well-ingrained and well-known.
The 1980s-1990s in the UK were magical decades for young people of melting pot popular culture, infusing the air with music and our sights with images from film, TV and 2-Tone - the Specials, UB40, the Selecter, the Beat - multiracial bands with an anti-racist ethos. Does anyone else remember 2-Tone's iconic black-and-white checkerboard symbol of racial unity? I mean, this stuff was the bread-and-butter of shows like Top of the Pops and Radio 1 when Harry was growing up and a drunken nuisance life-and-soul of the party.
Everyone knew about racial discrimination then in the UK, and knew that racial insults were wrong. Then came the time of transition from Thatcher-Major to Blair-New Labour, a time of new millennial hopes, Cool Britannia, and - ironically - Diana's funeral and pop-canonisation. Harry had opportunities to visit African nations and other countries, and to meet people from other cultures, experiences of which most teens and young adults could only dream. He was well-educated, with a cultured father and grandmother who knew the law and about social proprieties. He would rather play the drunken, racist thug to punish them for their existence and his not being Number #1 - that was his conscious choice. He grew to enjoy it.
Harry Windsor is now 41, coming up 42. He won't change.