Before answering, without being a student of the subsequent literature (I just read the two papers relatively thoroughly, but I assume some of the 2,000+ papers citing Putnam are not completely worthless), I would say Putnam's findings lend themselves to critiques of the type I have made but, well, better done.
To me, it tracks descriptively that "diversity" would correlate with lower average "social capital" (or a lesser average degree of "social establishment" in a local community).
A lot of it has to do with the historical formation of concepts of "diversity", a slippery concept the changes of which Putnam to his credit discusses more than the authors of the other paper. His article is framed in a relatively benevolent way, though I think it disguises a tendency to put forward problematic claims.
The drivers of increased human movement have everywhere included processes, instruments and events such as borders, warfare, religious oppression, human warehousing or "transportation", slavery, citizenship, empires and colonisation, and so on. Not to mention the simple possibility of actually moving from place to place that industrialisation has enabled via transportation. Unfortunately a narrower economic and quantitative lens doesn't provide a useful account of these.
It is important also to remember that none of these concepts are transhistorical. Even the concept of the "social individual" is a relatively new thing and isn't the same everywhere.
Over longer periods than Putnam or Alesina and La Ferrara consider, social establishment seems to go along with an erasure of historical forms of "diversity", such as the historical demographic and religious divide between British and Irish migrants to Australia I mentioned. It would not even be applied in census-taking these days.
In the medium term, those communities measured as "diverse" according to some living population taxonomy are to a greater extent those in which increased human movement has occurred more recently, or is ongoing. As in Putnam's study, they are nearly all urban areas adjacent to specific moments of increased economic opportunity. That all coincides with a normal understanding of groups of people everywhere tending to be rationally motivated by their material interests.
While Putnam controls for measured economic status and income using census data, as well as such factors as longevity in the community, he can't control for how people speculate on these things when they take life chances.
So while I find the data Putnam presents interesting (though as mentioned the order bothers me), like you I find it all (probably?) amenable to other explanations than his conclusion "diversity" "triggers" "hunkering down". For me that word "triggers" is a bit of a problem that can't be sustained just by doing some defensible statistical work.
Alesina and La Ferrara's paper I find rather circular, but if diversity is taken as a political given, it seems if anything to put forward a very strong argument against racism. Which goes to the policy question you raise …
Far from praising "official multiculturalism" as a process of universal integration and tolerance, critical race theorists will often mount of it as it is found in nations like "Australia", which a certain kind of politician will sometimes refer to as "the most successful multicultural society in the world".
From a certain point of view, multiculturalism seems like an institutional shibboleth that, as Alesina and La Ferrara put it in a somewhat sinister way, serves to "manage diversity" while perpetuating some sort of uncodified racial order.
To me this claim is supported when nations that have large populations of "othered" citizens such as Muslims in France or "Australia", people who may have been in the country for generations but are still subject to governments and representatives who declare "no Muslims should migrate here", or retrospectively suggest it was a mistake to let prior groups of Muslim migrants become citizens.
In my opinion, this kind of rhetoric functions as a domestic political assurance to some (white) people that their modest socioeconomic advantages will be helped along by supporting this or that public figure. In BLACK RECONSTRUCTION for instance, WEB Du Bois gives a compelling account of the political psychology of the so-called "poor whites" in the South of the United States which remains topical in the era of Trump.
"Ethnicity" (or race) can be seen to be constructed in very different ways in different times for different peoples in different places, for reasons that can usually be traced back to the demands of capital in a convincing way. Patrick Wolfe's book TRACES OF HISTORY is an excellent resource for understanding, for instance, the differences between processes of colonial racialisation that aimed at genocidal elimination, such as that between settlers to "Australia" and First Nations people on this continent, and processes that aimed at labour exploitation (such as that between the white plantocracy and African slaves in the South of the USA).
These differences have powerful echoes in the present, and in history they can be seen to line up in retrospectively quite obvious ways in such instruments as "blood quantum" laws. In short, not all constructions of racial difference are equal, and any model of the socioeconomic effects of "diversity" that proposes simpler theories of in-group and out-group relations without taking this into account seems flawed to me.
