I'd say one of the fundamental lessons of field and Galois theory is that they're not intrinsically special. They're just easier and more appealing to write down. (For the most part, anyway. They're a little bit special in some subspecialties for some technical reasons that are hard to explain.)
One reason to focus on them: when you tell students that x^5 - x - 1 = 0 can't be solved by radicals, that "even if God told you the answer, you would have no way to write it down", this is easy to understand and a powerful motivator for the theory. It's a nice application which is not fundamental, but which definitely shows that the theory has legs.
If you want to know which polynomials are solvable by Newton's method? All of them. It illustrates that Newton's method is extremely useful, but the answer itself is not exactly interesting.
> "even if God told you the answer, you would have no way to write it down"
But isn't this also true for generic quadratics/cubics too? Like the solution to x^3-2=0 is cubed_root_of(2), so it seems we can "write it down". But what is the definition of cubed_root_of(2)? Well, it's the positive solution to x^3-2=0...
When I say that a fundamental lesson of field theory is that radicals are not really special, this is what I mean. You are thinking in a more sophisticated way than most newcomers to the subject.
I feel there is an interesting follow-up problem here. The polynomials x^n+a=0 are used to define the "radicals" which is a family of functions F_n such that F_n(a) = real nth root of a = real solution of x^n + a = 0. Using these radicals you can solve all quadratics, cubics and quintics.
Now take another collection of unsolvable polynomials; your example was x^5 - x - 1 = 0 and maybe parameterize that in some way such that these polynomials are unsolvable. This gives us another family of functions G_n. What if we allow the G_n's to be used in our solutions? Can we solve all quintics this way (for example)?
I'm not sure I understand your question exactly, but I am fairly certain that not all quintic fields can be solved by the combination of (1) radicals, i.e. taking roots of x^n - 1, and (2) taking roots of x^5 - x - 1. I don't have a proof in mind at the moment, but I speculate it's not too terribly difficult to prove.
If I'm correct, then the proof would almost certainly use Galois theory!
He is not wholly wrong, while not solvable in the general case by normal radicals, there is a family of functions, a special "radical" as he said, the Bring radical that solves the quintic generally. Of course as said its not a 5th root, but the solution to a certain family of quintics.
One reason to focus on them: when you tell students that x^5 - x - 1 = 0 can't be solved by radicals, that "even if God told you the answer, you would have no way to write it down", this is easy to understand and a powerful motivator for the theory. It's a nice application which is not fundamental, but which definitely shows that the theory has legs.
If you want to know which polynomials are solvable by Newton's method? All of them. It illustrates that Newton's method is extremely useful, but the answer itself is not exactly interesting.