Why skilled workers come to Germany and then leave again

As a native German, I actually have difficulties with concepts like "identifying as [nationality]", "feeling like [nationality]" or "naturalization". I really would say these are concepts that US-Americans (or people who were "shaped" by US mentality) seem to deeply care about, but Germans very typically don't.

So, my opinion/advice is: she should simply abandon such concepts ("identifying as [nationality]", "feeling like [nationality]", "naturalization") that, as a native German, are simply far away from the mentality that I observe in daily life.

Don't forget that a unified Germany was a concept that only involved in the 19th century, so "Germany" is more of a somewhat "synthetic" unification of various historical, and very different, federal states where the unifying element is rather what is now considered to be a shared language, ethicity, culture and history.

With this in mind, the advice should be obvious:

This woman should concentrate on getting really good in German, and learn about the more than 1000 years of (what is now German) culture and history, and additionally learn about the laws and rules to survive daily life. Otherwise, she should live her life.

What she should not do, is caring about what "identifying as" or "feeling like a" German means - she should put this out of her mind, since modern Germany is a very synthetic unification of what were historically very different sovereign nations that share what is now considered to be a common language, ethicity, culture and history.

It's a serious question because in an ideal (IMHO) society, people can have full and satisfying lives with security and family without becoming a CEO. In the US, for example, there's an obsession with "getting ahead" but, by definition, only so many people can get ahead. And why do they want to? Because, at least in part, a basic job in insufficient to make ends meet in most cases now. This is a form of coercion.

This is orthogonal to the issue of German social inclusion and forms of xenophobia (eg in the housing applications you mention).

Personally I'd rather in a society where everyone's needs are met and it's not a race against a rising tide where only 20% of the population are above it.

That's not true, but it is a commonly shared myth. I've taken and passed C2 with the highest mark in every category (I moved here when I was a young teen, wanted to know if I would pass it after hearing years of people saying things like you're saying).

Most Germans would easily pass C2, although I think they'd have to be well-read/possibly university educated to get high scores (mostly need to be able to read quickly, give a semi-structured presentation and write a persuasive essay).

For what it's worth, I could run linguistic laps around all the other test takers there that day, and I assume at least some of them passed.