That is patently false. Especially the Western and Northern EU countries have a massive influx of high-skill workers (doctors, programmers, engineers) from both Eastern EU countries (which will dry up sooner rather than later, to be fair, with terrible population growth already happening for more than a decade in places like Romania), but also from neighboring countries - Turkey being the largest source.
Even the so hated "refugee crisis" from Syria saw mostly middle-class Syrians moving to Europe (putting it in fear quotes since the whole of the EU had fewer refugees than Turkey alone to accept).
>Especially the Western and Northern EU countries have a massive influx of high-skill workers (doctors, programmers, engineers) from both Eastern EU countries
That's mostly false. I haven't met a single programmer from Romania in Austria or any dev from turkey who wishes to emigrate here. Non-EU skilled people I met all want to move to low tax high salary places like Switzerland, UK, Netherlands. Having an open borders immigration policy is discouraging for those with skills who can afford to shop around for the better option vs economic refugees who shop around for the country with the most welfare.
If I look at the immigration statistics here, most immigrants are refugees from Africa, Middle East, and East Asia, not doctors and engineers with visas, those are only a tiny minority of the total immigration.
You may not have met them, and Austria may not be very attractive for this type of immigration, but there are significant populations of them. Remember that BioNTech (who created the European Corona vaccine) was in fact founded and is operated by Turkish immigrants to Germany.
> Non-EU skilled people I met all want to move to low tax high salary places like Switzerland, UK, Netherlands.
The Netherlands at least is still part of the EU. You'll also find plenty of high-skilled migrants in Ireland as well, and France and Germany also attract quite a few.
Note that language concerns and attitudes towards foreigners, especially foreign workers, are a huge part of how many people come to work in a particular country. Ireland and the UK have a huge advantage here purely by virtue of speaking English, as does France for the populations in North Africa. The Netherlands are very open to using English in business, even though their own language is quite obscure. German is less internationally spoke than English or French, but it is not that obscure either.
Even the so hated "refugee crisis" from Syria saw mostly middle-class Syrians moving to Europe (putting it in fear quotes since the whole of the EU had fewer refugees than Turkey alone to accept).