The only European Uni I knew was oxford, and I had no idea that I could save money on a similar education in Europe. In the first place I just assumed I would need to know a foreign language for any non-UK European school.
Heh, yes. A long time ago there was this guide aimed at foreign students coming to study in the US. They had a warning: if the school pitches itself at foreigners it does not have sufficient domestic students. The prospective customer should ponder long and hard about the motive of the place.
(The motive may not always be nefarious. Back then, the Classics departments of Oxford and Cambridge did pitch themselves at German school leavers because in England Latin instruction was plainly worthless. In Germany it was actually at an adequate level.)
It's not in the interest of any European government or university to get flooded by American (or any other foreign, non-EU) students. The only reason some German states haven't introduced a fee for non-EU residents yet is probably that their number is not that high yet. But some other German states have already set up such extra fees. Some other countries place fees on English-language programmes, while those in the local language remain free.
Some ignorant american would just drink, cost money and move away at end.
They wouldn't, but American universities do and it works. Thus the answer to GPs conundrum of why don't more students consider school in Europe.