I suspect answers to this question is going to be strongly affected by one's perception of the pros and cons of multiculturalism vs nationalism. Still, considering that many governments put in place policies promoting one or the other, it seems like a useful policy question.
First, you remark, correctly, that some celebrate multiculturalism, while others bemoan the loss of the nation's identity.
The first group calls multiculturalism beneficial without bothering much to get into specifics why. I.e. "multiculturalism is good because warm and fuzzy and nice".
So, as I understand the question, you're asking: "What are, instead, the utilitarian benefits, if any, of multiculturalism?"
What are the utilitarian benefits, if any, to uni-cultural national identity?
As far as I'm concerned, the second group, those regretting national identity loss are equally unspecific as to the actual, tangible, gains of "cultural cohesiveness".
If we accept the first question, then the flip side "Why should I care if my ethno-nationalist identity gets diluted?" also bears asking. I haven't seen too many good answers to that question either.
That's not your question, but actual data-based answers to it would provide a valid yardstick to evaluate claims about multiculturalism.
i.e. why hold the first group to a higher standard than the second?
Second, probable lessening of negative outcomes.
Before one gets into alleged positive, active benefits of multiculturalism, one might also look at what it probably lessens:
ethnic tensions between groups
economic marginalization of immigrant groups. Having people stuck in low-paying jobs will forego their potential contributions and tax revenues. Many Western European countries, France being the one I am most familiar with, suffer much higher rates of poverty and joblessness among "immigrants", extending to 2nd and 3rd generation, and affecting people who have the citizenship but are not members of the "traditional historic citizenry" .
Third, is it really an either/or?
Can a country not be both multicultural and take pride in its feeling of shared nationhood?
Certainly it is possible to feel dislike for both sides in this type of debate: some people dislike both white nationalism and, to take an example, the overextension of the initial legal doctrine of critical race theory into an overarching catchall "blame culture".
So, is it not possible to take pride in your nation and its multiculturalism?
OK, enough of the preambles, let's look at some arguable specific benefits of multiculturalism.
Food and arts
Would the average UK citizen really, really, want to roll back the clock to days of traditional English cooking? No more curries? Ditto in Canada, it would suck for us.
Even France, with its wonderful traditional cooking, benefited from introducing foods from cuisines which better catered to quick casual eating: pizzas, couscous. And, yes, "Le Burger" too.
Music. Aside from chamber music, I'd be bummed if you took out all the modern music genres that descended from rock and forced me to listen to music from "my Caucasian roots". And rock descended from Black American music. To say nothing of jazz. Or rap.
Paintings. The Impressionists were heavily influenced by Japanese art in the late 1800s. Would we better off if the clock had stopped at the Romantics?
These aren't just feel-good benefits. Cultural exports from the music industry has surely contributed massively to the UK and the US's trade balance over time.
A country which is known to be unwelcoming to immigrants will be an acceptable destination to those fleeing real and immediate threats or abject poverty. The trouble is you get both skilled and unskilled immigrants that way.
On the other hand, a country known to be welcoming, or at least granting opportunities to, immigrants will also be desirable to immigrants who have more agency in where they want to settle and those will tend to be more skilled: doctors, engineers, business people.
That's true for Canada, even though we sadly have too many engineers or architects immigrants driving taxis and other underemployed roles. That's also true for the US, when compared to Europe, even though the US gets a lot of bad press about racism.
Since multiculturalism is often seen as positive, being known for it can make one look more virtuous. On the international stage, Canada's virtues in that domain helps lessen awareness of unsavoury behavior like discrimination against natives or excessive emissions per capita.
Multiculturalism doesn't have to mean accepting everything.
Some cultures have rather regressive views on subjects like gender equality or homosexuality. Are we obliged to accept those aspects? (keeping in mind that our own fairly recent history, from sometimes belated universal vote for women or things like Alan Turing's treatment leave much to be desired)
We are not. There is nothing wrong with rejecting those particular facets of foreign cultures. But being respectful on the rest gives better "traction" when engaging with a community to curb some of its members.
Openness to new ideas.
Both the Ottoman Empire and China, at one point the top dogs in their time and geographic neighborhood, started sliding into irrelevance when they convinced themselves that they were the be-all of civilization and deliberately cut themselves off from foreign influences. The same thing happened to Japan, though it switched paths extremely quickly. Intellectual and economic autarky are both risky paths for a country to engage in.
In our modern world, Westerners will not be the only, or even primary, innovators in industrial, technological and economic fields. Kanban and just-in-time manufacturing to take examples, have revolutionized how we make cars and other goods, but came from Japan. Ditto for taking in foreign scientists (cribbed from Meriton).
If we withdraw onto ourselves, rather than accept immigration and ideas from various countries, would we risk missing out on the next new thing? Would we not risk going the way of the Ottomans or 1500's China?
A culture which is confident does not fear the influence of others.
Overfocus on today's differences ignores how past ones worked out.
Once upon a time, "others" in European countries were considered a big problem too. You'd have to look no further than the first waves of Irish and Italian immigrants to the US and their reception. Or how Spaniards and Portuguese were perceived in France. Modest Proposal was satire, but not conjured out of thin air.
Guess what, it all worked out in the end, though it seemed dramatic at the time. The newcomers ended up strengthening their host countries.
How well does the alternative work?
Do countries that advocate immigrants dropping their culture and fully adopting their new host country's instead do better? Contrasting France and Canada, both of which I have lived in, I have a definite opinion.
While a country does not HAVE to accept immigrants, it SHOULD be nice to them once they are accepted in. That's both the decent, and utilitarian, way to proceed.