> Its cultural diversity is a plus for most people
Cultural diversity is not a draw. Africa, Latin America, and Central Asia are all culturally and ethnically diverse. No one moves to any of these places.
The primary draw of a city like London is economic prosperity, which is ironically usually only made possible by ethnic homogeneity. This is the case in Britain’s former colonies (USA, Canada, Australia, New Zealand), China, Saudi Arabia, etc.
Target cities become “culturally diverse” due to the arrival of migrant labor. The migrant labor itself is not seeking this diversity out for its own sake. New migrants have their movements facilitated by networks of already-landed migrants, who provide knowledge of the immigration process, employment opportunities, and material assistance to their coethnics.
These people are not moving to London because they can find people like themselves there (there are already plenty in their country of origin), nor are they moving to London because they want to experience other cultures (this was a form of conspicuous consumption that went out of fashion years ago). At best you could say they hope that the relative ethnic heterogeneity will distract from their own foreignness, but that still doesn’t amount to being drawn to diversity.