If they had a weirdly large supply you can be guaranteed an army of recruiters from FB, Amazon, Google et al would be lining up at campuses the way they do in India.
The only limitation these firms face in growing faster is getting their hands on talent.
Bullshit. Tech companies are not competing for worldwide talents. The companies you quote have at most 3 offices in Europe: London, Zurich, Dublin. The first thing a recruiter will tell you is to abandon your life and your family.
Google has offices in almost every countries but they are administrative offices with almost no employees. If you're a tech worker, you can only be in one of the very few tech locations.
They do have tech workers in multiple locations in Germany. At least one of them is almost only tech, but small.
But you are right, that those are not big major tech offices like the three mentioned. So most engineers in Europe would still be working out of London, Zurich or Dublin.
And also leaving your language, your currency, your bank, your culture, the entertainment you know (movies, culture, songs), the shops and brands you are used to...
The only limitation these firms face in growing faster is getting their hands on talent.