Current undergrad taking math courses at one of the colleges mentioned. I would say that what you suggested is definitely part of the problem. The other part is that even the good* programs in high school are very different in structure from university math courses. There are very few study skills developed because most of the focus is on remembering the testing algorithm and does not necessarily include the rigor required to fully understand mathematics.

*good in this context means AP level. This is not because AP is the correct measurement, it is because AP and the like is the measurement used to judge high schools