There is a saying about London and the US:
'People go to America to become Americans. People go to London to be themselves'
Hmmm, sounds like a saying that was coined in… London. Honestly, if you think that Americans are any less 'themselves' or varied than Londoners then you have obviously never been to NY or LA or…etc etc.
Having lived in both London and the States, here's my 2 cents/tuppence:
All that the melting pot/salad bowl 'American' approach is saying is: There should be an element of self-identification on the part of the immigrant as an American and cherish certain civic/constitutional values. They are free to keep any non-conflicting elements of their original culture (be it the cuisine or whatever).
When the British approach works, it is because there is some self-identification on the part of the immigrant as being part of British society and adherence to British laws/democracy etc. Where it doesn't work, and alienation develops is often because those ('melting bowl') elements are missing.
Note: this isn't saying that it always works in the USA (there are immigration successes and failures in both countries) but I feel that the US approach, by emphasizing a shared bond of citizenry, is probably superior for a healthy democratic society.