I am not a graduate student (yet), and don't really know about what first-year grad courses are like, but I don't think that Galois theory should be dropped because "it takes too long to develop", at least not the finite theory.
The only reason why (finite) Galois theory seems to require a lot of time to develop is (I believe) that the standard (I'm inferring from textbooks and course notes I have read) development is backwards. All questions of constructing extension fields (which are the only place where you need ring theory) can be tacitly omitted (or given as homework in a ring theory module), while extending field automorphisms, and the notions of splitting and separable extensions, can be motivated and illustrated by determining the Galois correspondence itself.
A brief outline of the lectures in my head is this. Suppose E is a field. The stabilizer of any group of automorphisms of E must be a subfield of E. Given F a subfield of E, what are necessary and sufficient conditions that F be the stabilizer of a subgroup of automorphisms of E? (why do we care? you're a grad student, that's why you care.)
Clearly, these automorphisms will belong to Aut(E/F), and one immediately derives the necessary conditions that any (irreducible) polynomial f with coefficients in F that has a root in E must have deg f distinct roots in E, since otherwise the product of the (x-r) where r ranges over the distinct roots of the polynomial in E will be fixed by Aut(E:F) and of lower degree than f, thus it will have some coefficient not in F and that coefficient will be fixed by Aut(E/F), which would mean that F is not the stabilizer of Aut(E/F).
The two ways in which polynomial can go bad, then, is that they don't split or have repeated roots, thus motivating the notions of split/normal and separable extensions.
From this perspective, the fundamental theorem of Galois theory says that for finite extensions, the conditions of splitting and separability are in fact sufficient. Here you develop the necessary tools of degree of extension fields, simple extensions, extending automorphisms of intermediate fields and (bless his soul) Emil Artin's linearly independent characters, and you're done (with finite extensions).
