The UK has had increased (net) immigration for decades, including a huge jump in the past few years[0]. The population grew commensurately, likely mostly due to immigration, since as you accurately point out the birth rate is low; there are roughly 13% more people in the UK now than 20 years ago. Why, then, has the GDP per capita flatlined in the same timeframe? Where is the growth that immigration is supposed to fuel?

> Immigration is the only thing fueling that population growth

This is a core neoliberal claim that smacks of "we've tried nothing and we're all out of ideas".

> The real argument should be about rates of cultural assimilation or acculturation

By all means, let's have that discussion! If immigrants were, by and large, able to acculturate well, there would be much less anti-immigration sentiment.

But the same neoliberalism that supported increased immigration to developed countries on these economic grounds also morphed into a dual prong of claiming that diversity is our greatest strength and that it's racist to expect immigrants to have to conform to their new host culture. Both attitudes are directly at odds with assimilation and acculturation.

Other wealth European nations that took in a lot of immigrants, mostly from grossly incompatible cultures, have seen concerning increases in violent crime and other things deleterious to social cohesion and quality of life.

> (See Japan and Russia)

Japan's population has been shrinking since 2010; its GDP per capita has been treading water for even longer, since the early 90s. Similar story for Russia, whose population has been flip-flopping from growing and shrinking for a few decades, and whose GDP per capita hasn't seen any sustained growth since the late 2000s.

So it's entirely possible to have a stagnant economy, whether or not you receive immigrants.

> Without larger younger, working age generations, social safety nets and productivity investments are difficult to finance

This is about the only unarguably true statement you made. A nation composed of too many nonworking people receiving transfer payments from working people cannot exist in that state for long.

[0]: https://www.statista.com/statistics/283287/net-migration-fig...

Aside from not being the only nation outside of Schengen, Schengen only implies freedom from borders checks. As an EU citizen you could still emigrate freely (or as freely as anywhere else in the EU) to a non-Schengen countries. I know I did.

"Uncontrolled" in this context doesn't mean people hopping the border unbeknownst to the authorities. It means immigration at a rate high enough that it's beyond the capacity of the host nation to integrate the newcomers to its culture.

A village of a thousand people can accept one additional person per year indefinitely with no material change to its way of life. But it cannot accept, say, a hundred a year, especially if the hundred immigrants come from a substantially different culture, without fundamentally changing the entire social fabric.

And when such massive changes to society occur without buy-in from a convincing majority of people?[0] Well, only idiots would wonder why people think it's out of control.

[0]: In the UK, Brexit made it clear that in fact the buy-in was in the other direction.