The theory of weights i am honestly not sure. It can be stated entirely in terms of Lie algebras. The theory of algebraic groups of course, but i think it’s only correct to say matrix exponentiation “leads to” that in a historical sense.
Jordan himself was working on questions related to Galois theory when he developed his normal form. He actually did it in positive characteristic first! But Weierstrass had in fact invented Jordan form a couple years earlier, while trying to classify bilinear forms over C. The connection to matrix exponentials comes a couple years later when somebody was trying to solve dy/dt = Ay with two variables.