I understand that the study of Boolean circuits and nonuniform complexity classes was introduced to (hopefully) prove separation of uniform complexity classes. In this sense, nonuniform computational models can be thought of as a secondary device in complexity theory, the main interest being the uniform model.
However, in algebraic complexity, the situation seems to have been the other way around. Algebraic nonuniform models including algebraic circuits were studied long before the uniform Blum-Shub-Smale machines were introduced. I am confused about this; since nonuniformity preceded uniformity in the algebraic setting, something about nonuniformity should have been unsatisfactory. But what was wrong with straight-line programs?
I'd be grateful if you could tell me why uniformity in algebraic complexity theory can be preferable.
EDIT: I realized that uniformity helps if you are discussing computability of algebraic problems.