Eh, you're being pretty harsh.
I know a few (not too many, but a few definitely) medical and dental students who have failed a class, and I don't think that they will graduate from school to become incompetent healthcare professionals.
Why people fail varies. It seems like you're assuming that people fail because are incompetent. Sometimes that is the case, but it is more likely that these people do study, but their grades are around the pass/fail border and they end up failing something at some point. When that's a case of a poor work ethic, that's an easy fix. Something I have seen that is unique to dental school is that some dental students are really slow to pick up hand skills and therefore fail a lab class in the beginning.
I definitely don't think that failing is par for the course, and I think everyone should strive to do the best that they possibly can. Most students will pass, however I wouldn't completely discount someone who failed a class. What is more important than whether that person failed a class or not is how that person picked him/herself up, remediated, and was able to improve so that he/she could pass the rest of his/her classes in order to graduate.
People who fail multiple times do get kicked out. I am sure of that. Standards are rigorous. I know for a fact that there is at least one school (not talking about NYU here) where someone doesn't need to fail a class to get kicked out.