Warning: I am basically just speculating, and not commenting with actual knowledge of the history.

I suspect that a lot of the nineteenth century work on determinants was motivated by invariant theory. Before Hilbert proved abstractly the finite generation of invariants, there was a small industry trying to explicitly compute invariants (for example the projective invariants of the action of $GL_2$ acting on binary forms) with the aim of giving a constructive proof that they were finitely generated.

These invariants often have a basis consisting of various sorts of determinantal expressions, and if you want to prove finite generation, you have to construct ways of taking certain determinantal expressions and writing them in terms of other determinantal expressions.

Within a decade or so of Hilbert's paper, people generally lost interest in constructive invariant theory. (After all, the abstract methods answered the most interesting questions and seemed much more likely than constructive methods to answer the most interesting remaining questions.)

I have wondered whether all the work on syzygies of determinantal varieties actually reduces to identities which were well known (to the right people) in the 19th century.