I guess that depends on which "classical" algorithmic problems he is referring to. The field of "algorithms" is like an infinite sea which will never be fully explored; there will always be more algorithmic problems for which no library is available.

For example, last year I worked on a software package which was intended to match clothing automatically to form outfits. The client was a fashion consultant; the software had to duplicate what he did in his day-to-day work, algorithmically. You won't find that one in "The Art of Computer Programming"! (Or in any library, for that matter.)

Depends on what you're doing. There are plenty of cases where, for example, the Java collections library doesn't quite do the trick for various reasons. E.g. the standard collections are very memory inefficient for storing primitive types, or the standard linked list implementation doesn't support all the operations you might want - e.g. being able to clone an iterator.