In my city we have medical students coming to study from all over Asia and Africa. They will become doctors, surgeons and other specialists. Many will use this as a stepping stone to stay in the EU. What you said is flat out false.
The more important stat is what the distirbution of immigrants vs natives working as doctors or nurses in the EU is compared to the distribution of migrants vs natives in the total population. I believe you will see that migrants are generally over-represented in Healthcare compared to their percentage of the total population, and similarly in certain other industries.
I have not been able to find very recent data, but here is a paper from 2007 [0], showing that the percentage of physicians working in the country at that time who were migrants either from inside or outside Europe was well over 5%, with significant outliers such as the UK (30%), Ireland (30%), Norway (15%). In Austria it was 5%, though most migrants were from other rich EU countries.
