we will automatically find that most of the useful research was done by people who found it interesting.

There is still a distinction between problems that were worked on exclusively because they were interesting, and researchers who worked on important problems that they found interesting. For example, presumably Cayley found systems of linear equations interesting. At the same time, he could provide strong practical justification for understanding the solutions of such systems; he did not have to appeal to interestingness to justify his work.

This is a different situation from the one in which someone decides to study matrix algebra because matrices seem like intrinsically interesting objects (which is also something that frequently happens in mathematics).