Here are some further comments, in community-wiki format. As earlier comments indicate, compact Lie groups and their maximal tori raise many questions. For instance, the real case is perhaps more basic (and also more familiar to most students from their linear algebra acquaintance with orthogonal matrices, etc.). Complexification mainly connects the subject with Lie algebra theory. But you'd clearly prefer to avoid exceptional types and such.
One possible approach is to rely on the most familiar examples: O(n) and SO(n). In these cases, there is the old theorem of Dieudonne, asserting that every group element is a product of at most n "symmetries". But since a reflection in this concrete setting has determinant -1, there is an extra step in passing to N(T)/T in the connected Lie group SO(n) when n is even.
Beyond matrix theory examples, a delicate issue in relating compact Lie groups and (finite) Coxeter groups is that not all of the latter finite groups are Weyl groups coming from conventional root systems, e.g., most dihedral groups. The classifications in Lie theory and Coxeter group theory are close but not identical. Maybe the optimal connection is seen in the context of work by Tits, usually involving BN-pair language. Coxeter himself was a geometer who studied especially finite and affine groups generated by reflections, but Bourbaki and Tits developed more of the Lie theoretic connection after early work of Witt while broadening the idea of "Coxeter group".
Good luck!