I will write from the perspective of an undergraduate student. I think one reason undergraduate math is not interesting is the technical jargon laid before undergraduate students in the learning process. Usually one had to understand why one has to comes up with such a particular method, a concept, or a theory to understand the subject. The utility of any math they learned usually can only be manifested by sample calculations or problems.
But there is something much greater than this. For a really bright first year math student, I would assume normal course material is insufficient, and textbook+lecture+office hour do not solve his or her problems. Nevertheless the technical tool available to him or her does not enable him or her to do real research. I remember when I was a high school student I asked a math graduate student if there are two different holomorphic structures on a four dimensional manifold. At that time I know some complex analysis and some differential manifolds, but they are not sufficient to give an answer to this. Even now I do not know a yes or no answer in a form I can understand. A really bright first year math student is very likely to ask strange questions he or she cannot solve, and it stuck him or her so much that he or she decided to quit math. Equally likely is he or she found Putnam problems too technical, problem solving too time consuming, and in the end learning math becomes mastering a machine. I still remember the days I was stucking with my problems in reading Hatcher, at that time I was serious considering of transfering to history. And I am hardly a bright math student.
The solution to such a problem should be to guide him or her to read some master's work. If he or she can understand how a working mathematican did to tackle a problem, developing a math concept, or expanding a theory, it would be very beneficial because it helps the student to achieve a higher level of mathematical maturity unavailable by normal curriculum. Abel once replied "I read the masters, not their pupils". Clearly a lot of past mathematical work had been discarded or being forgotten, yet the professor can always choose something relatively more readable. If this is not possible, the professor should at least devise an individualized class with good supplementary reading.
What I never understand is why the professor will adopt a useless undergraduate level textbook in his or her class instead of reading good math articles, or even lecture notes. I would hope the professor can pointed out where the material is insufficient, where the results had been updated, and where better strategies had been developed. A good list of sample problems coupled with good mathematical reading can change people. Sadly I did not had a chance to have such a class in US. As a result I had to go to Moscow, where I rediscovered math. But I hope future students would have a better fate.