A big part of why I think my hypothesis would work is that I believe we have these shared values in the US.
The problem is that these shared values have been obscured by disagreement on how to attain them, and by focusing on group identities that mark us all as "Other" to someone else.
I like your idea. At worst, you'd find out sooner or later if these shared values are a myth. Getting to know others is never a mistake.
I met plenty of different strangers from around the world through Hospitality Club (the concept later copied by CouchSurfing) and never had a bad experience. Many of my friends and family thought it was dangerous inviting strangers into your home. And maybe today it is, I'm not sure. But back then practically everyone signed up there had the same open-minded spirit, they wanted to learn about people in other countries and further intercultural understanding and friendship. I've met lots of people from very different walks of life, people I might have never otherwise talked to. This is positive diversity, when there's a shared value and people can unite around it and trust each other.
To give a negative example many people in Western societies, in their strive to not be racist, have bought into the false belief that all cultures are equal, that you're assigned one by birth and that you're not allowed to criticize the culture of "other" people. Which ironically in itself is a racist idea. This in turn has lead to political movements that consider it taboo discussing certain problems connected with topics like religion, culture and ethnicity. And it's given birth to the kind of laissez-faire multiculturalism where problems never get addressed and crime, mistrust and conflict are rampant. Because people either get divided among identity lines like you mention, or unchecked immigration gives these societies the death blow (the Western European model).
In the early 20th century, “multiculturalism” meant northern, southern and eastern white Europeans living together. In the documentary Australia in Colour, vintage TV advertisements portray immigrant ships containing northern and eastern white europeans as being “good for Australia” — a point of controversy at the time. Tellingly, what modern Australians would consider actually multicultural, e.g. integrating with E.Asians or Australian Aboriginals, was utterly unthinkable.
Yet in Australia and the United States today, the zeitgeist is that we’ve “always been multicultural”. This is clearly only superficially true: multiculturalism was only recently redefined to mean the integration of non-white europeans into predominantly white european societies. In the documentary, they explain following World War II, it was feared that Australia would grow weaker than its Asian competitors oweing to Australia’s low population, which many in government thought made them susceptible to capture by aggressor nations. This, AFAICT, was the real impetus for the Australian government becoming increasingly accepting of Asian immigrants.
And more recently, during the pandemic, the Australian housing sector has been calling for more immigration. Apparently, immigration powers much of the Australian housing sector [1]:
> A fall in migrants during the pandemic is causing a sharp drop in housing demand, with the sector urging the government to create a migration plan and extend HomeBuilder incentives.
Modern multiculturalism seems to be primarily based on economic and militaristic concerns, and is in no way based on a desire for social cohesion. See also: One Billion Americans [2]. Some say the US should strive to increase its population size to one billion for more or less the exact same reasons modern day multiculturalism came about. These people were never concerned with social cohesion: they’re entirely concerned with militaristic and economic might.
The problem is that these shared values have been obscured by disagreement on how to attain them, and by focusing on group identities that mark us all as "Other" to someone else.