The American system is fairly broken, true. But it's important to remember that the vast majority of people (a number around 90%) who go to university (even expensive private ones) here in the U.S. manage to navigate this system and go on to productive, non-bankrupt lives after graduation.
It would be better if it was 100% of course, but it's not like most people leave school and have to fight a desperate battle against debtors. For most people it's just a temporary drag on their income that gets paid off relatively quickly if they earn enough to pay over the minimum.
The reason it sort of quasi-works in the U.S., but seems inconceivable in Europe, is likely the differences in the labor markets. In general the U.S. doesn't have the same kinds of problems (or as severe) with youth employment and underemployment. For example, during normal economic periods, youth unemployment in the U.S. is around 13-15%, during the recession we considered it to be a major crisis when it went to 20%. By contrast the EU youth employment rates (not unemployment) sits at around 30% (with Germany being a notable exception at ~8%).
Furthermore, it's generally considered acceptable in the U.S. to job-hop for higher salaries every year or two, meaning that your earning potential 5 and 10 years out of college will be much higher than directly after graduation, further improving your ability to pay your loan off.
In other words, it's more possible in the U.S. to take on loans for education, and have a reasonable expectation of graduating out of college into a full-time job, where you can further expect to have increased your earning ability and thus pay the loans off.