Maybe I should try to defend myself, or at least the self I was four decades ago when I improvised my graduate text. But first I should disclaim any originality in the proof of Weyl's theorem, which I drew from Bourbaki (who didn't invent it either). I was at the time far from being an anti-Bourbaki person and in fact actually associated with some of them. But in my limited experience teaching real-life US graduate students, I was well aware that it was unrealistic to build up Lie algebras abstractly in the Bourbaki style. In particular, I didn't want to talk about the universal enveloping algebra (and its center) until that was really necessary. At the same time, I really felt that it was useful to talk about Weyl's theorem earlier than usual since I was emphasizing representation theory rather than general structure theory.
Indeed, as a later addition to my book indicated, Victor Kac demonstrated in the mid-1970s the value of working with just a single Casimir-type operator in his approach to affine Lie algebras and what came to be known as the Weyl-Kac character formula.
Anyway, it's clear that the approach of Weyl imitating the classical treatment of complete reducibility for finite groups is the most natural, but for this you have to be in the framework of compact Lie groups and invarioant integration. There are advantages to working with Lie algebras directly in a purely algebraic framework, though of course some of the ideas lose their original group-theoretic motivation.
For me the algebraic proof was a good illustration of the power of slightly abstract algebraic thinking, especially for students not previously exposed to such proofs. But the main motivation for introducing the Casimir element would be to have an "invariant" commuting operator (essentially living in the undefined universal enveloping algebra) relative to the given finite dimensional representation. Where this operator comes from is another question, since Casimir's own work had a physics origin. Whether or not it helps to provide an intrinsic definition for the operator is another question, but for that there is an earlier MO question as well: 52587 along with my earlier historical question 41150.