The Lie of integration has so much room to be analyzed, tore down, and debunked. Not only does integration ask someone to strip themselves of their cultural identity just to conform to the societal norm but often it asks the same individual to condemn their original identities and views. Majority cultures, especially European cultures but most especially the British, love to flaunt how 'accepting' and 'educated' they are of/on the diaspora of minority (conquered) cultures that were present in the British Empire, and then in the same breath condemn multiculturalism and promote the systematic condemnation and destruction of these cultures within Britain. It is not just a 'cultural conversion' that is pressed onto melancholic migrants but the active condemnation of their mother culture, once 'cultural conversion' has been completed society seeks to implicate the melancholic migrant in the destruction of their old identities. All of this, just to never truly be accepted by the Majority as a "proper British bloke".
In a far less literal sense than the example of Black American Veterans that you gave, "Integration" seeks to enlist the migrant/minority population in a war against their own communities. "Multiculturalism" is dead only because the societal Majority is too lazy or ignorant to seek an understanding with its minority populations. Every mention of "seeking understanding through 'shared values'" makes me laugh and cringe, knowing that the real intention behind those words is 'British values'.