The ironic thing about the whole "integration" thing is that Westerners tend to be the worst at integrating. Even in conservative Muslim countries they form their own expat communities where they somehow manage to get authorities to turn a blind eye to their cultural practices like drinking and wearing revealing clothing.
And I mean that's not necessarily a bad thing. Dubai has been liberalizing basically to attract expats. And now Kuwait is doing the same. There are rumours that Kuwait wants to build the first bar/night club. I dunno when/if that's going to happen, but there's definitely a huge push lately influenced by Dubai to develop the country and make it more appealing.
Societies develop by integrating other cultures. As you said, other cultures have a lot we can value too.
Yea, I have read about the whole expats thing, in the sense that the English are never immigrants, they are expats.
On account of being so special you know.
It's an interesting question. Considering the rise of segregated and divided communities and a growing us vs them mentality, do we need something to unite us as fellow citizens?
We do, but honestly the word British to me is already loaded with nationalism, and I don't think nationalism is the way forward. I also see it as very white to be fair, a position I can't even occupy.
Walking along Church Street off Edgware Road on most days, but especially market days will show you that people from a wide variety of ethnic backgrounds, races and cultures are indeed united by their shared experience living and working together. This morning I saw a Moroccan store holder helping and greeting his English neighbours running a fish stall and on the otherside a veg stall. They make and laugh together. Local characters shuffle by and say hi and shake hand or touch fists. English traders joke with a familiar hijabi customer. Polish and Iraqi shopkeepers sharing tea and olives. A niqabi stoops to pick up something a elderly lady in a wheelchair dropped. And so on… I love Church Street. Someone should do a documentary about it. It is British values in action. Tolerance. Cooperation. Humour. Hard work ethic. Helping others. Inclusiveness. Generosity. Etc... Of course at the same time these are universal values, but here they are displayed by people in the uk who share a local connection.
It's funny that you say that Hass, you know I have lived near there, run through its streets as a kid and a teen, attended weekend Arabic classes there, I still have family there, and I see it differently. Like that awesomeness you see, to me is a veneer, under which a lot of shit is happening.
In the cutest way possible, your description of it, reminds me of this song and video:
https://www.youtube.com/watch?v=wmYT79tPvLg 
I did my dissertation on racism amongst the Moroccan community, and interviewed people from there, and to be fair was confronted with some hideous racism and tribalism too. My step mum, I can see her easily having banter with a Polish market person, and then get home and saying some god awful things.
Maybe I am just way more pessimistic than you.
Topic: British Values