Galois theory is applicable to fields which are neither subfields nor extensions of $\mathbb{R}$ or $\mathbb{C}$, so that's one reason why one should not expect to be able to translate a Galois-theoretic proof into a real/complex-analytic proof.
The main problem with using analytical tools to solve the problem of the quintic is that the tools of analysis are not tuned specifically to handle algebraic functions. Sure, you need complex analysis to prove that every polynomial with complex coefficients has a complex root (assuming you are defining $\mathbb{C}$ analytically), and with a little more effort you could perhaps even prove that the roots, in some sense, vary continuously with the coefficients. But the question is, do they vary in an algebraic way? Here complex analysis falls down, because there isn't (as far as I know) an easy analytical way to distinguish between an algebraic function and a non-algebraic function. (What's the essential analytical difference between, say, $\sqrt{x}$, $x^3$, $\frac{1}{1+x}$ and $\log x$, $\exp x$, or $\tan x$?)
On the other hand, Galois theory enables us to distinguish between, say, $e$ and $\pi$ from $\sqrt[3]{2}$ and $\frac{1+\sqrt{5}}{2}$, and most importantly for the problem of the quintic, between the unique real root of $x^5 + 23 = 0$ (which can be written in terms of radicals) and the unique real root of $x^5 - 15 x + 23 = 0$ (which cannot). The key is the group of automorphisms of the field extension that the roots of these equations lie in, and this is an inherently algebraic concept.