Here are a few extended comments. First, it's always desirable to re-examine basic material as more of it accumulates and makes the research frontier look impossibly remote to students. The handy book What Every Young Mathematician Should Know (But Didn't Learn in Kindergarten) gets harder to write every year.
It's never easy to say what ingenious new approaches are possible, but the question in the header has already been tackled (unsuccessfully) in a moderately paced textbook by Karin Erdmann and her former student Mark Wildon Introduction to Lie Algebras (Springer 2006, reprinted with some corrections in softcover format). The authors maintain a list of corrections to both versions, including a further correction to Theorem 9.16 on preservation of Jordan decomposition. They tried to avoid Weyl's theorem on complete reducibility but tripped over a hidden obstacle.
As this cautionary example indicates, it is tempting to simplify proofs but not always easy. The rigorous approach taken by Bourbaki (and Serre) to such matters is reliable though not always user-friendly. In any case, semisimple Lie algebras can be studied over an algebraically closed field of characeristic 0 using just some linear algebra and basic abstract algebra. It's not at all necessary, except for motivation, to deal with complex coefficients, Lie groups, or linear algebraic groups. Even so, there are sophisticated arguments including Cartan's criterion for solvability which seem hard to avoid.
Historically the classification of simple Lie algebras over a field such as $\mathbb{C}$ doesn't involve notions like Chevalley-Jordan decomposition or Weyl's complete reduciblity theorem. It's at least partly a matter of taste which approach to take, but Casimir operators and such do come up naturally if one gets into representation theory. My main concern about restructuring the foundations is that it should be done rigorously and in a way that doesn't force students to go back and start over again if they decide to pursue the subject further.
P.S. Some other relevant questions on MO can be found by searching "Jordan decomposition".