On the contrary, moving out to establish a nuclear household is the common pattern west of the Hajnal line[0], along with later marriage, and the lower fertility which goes with it.
This has been true for at least a thousand years. America was largely settled by people from this folkway.
The influence that this pattern, and the bans on cousin marriage (out to fifth cousins in some cases, but second cousin marriage was commonly forbidden), has had on what we sometimes call WEIRD people, is interesting, if a bit off topic.
That link says nothing about multigenerational households.
At least in the early 20th century, multigenerational Irish, German and Italian immigrant households in New York were all fairly common, so I find it hard to square with what you're saying.
I followed one of the links to this very closely related article that does go into some more detail, though the actual link to marriage age is somewhat glossed over.
That is a very curious proposition, even though it only focuses on marriage age. I imagine it may have been true at some point (1965 when Hajnal looked at it), but countries of the former Yugoslavia have been having a net decline in the population since at least the 80s. I rarely meet anyone from those years (I was born in 1983) who has more than one sibling in Serbia, Montenegro, Croatia, Macedonia or Bosnia (haven't met many people from Slovenia, but they are probably even worse off). And many of my primary/high school friends are an only child too.
I attribute that to widely available "free" University education (women were equally represented in student populations, if not at 50%, at 45% for sure, though "citation missing") soon after the World War 2, housing availability, etc.
Knowing some internal details like the fact that Kosovo probably had highest fertility in Serbia until it split off, the situation in 70s probably mirrored the one from countries West of the line.
The pattern Hajnal notices more likely corresponds to migration into cities (move from agriculture to industrialization and now services) and wider availability of education and employment to everyone, but in particular women (which happened in Communist Yugoslavia) and rising standards of living, rather than historical cultural traits as he claims in his paper (though I found only an abstract).
With more effort and closer look, I am sure that all of this could have been disputed even in 60s when the paper was published.
And interestingly, more recent data suggests that West of the line now has a higher fertility than the East in Europe.
[0]: https://en.wikipedia.org/wiki/Hajnal_line
This has been true for at least a thousand years. America was largely settled by people from this folkway.
The influence that this pattern, and the bans on cousin marriage (out to fifth cousins in some cases, but second cousin marriage was commonly forbidden), has had on what we sometimes call WEIRD people, is interesting, if a bit off topic.