Great observation.
The American Math Monthly is a journal by college professors for college professors. The readership is expected to be familiar with the topic. Groundbreaking results are published elsewhere. Whew. Unfortunately, neither is AMM a place for professors to summarize the contents of a 14-week course for adult learners.
Let's use your observation to illuminate 2 abuses of notation that happen all the time between those in the know.
The first abuse is not explicitly calling out that the coefficients a_i are restricted. The polynomial of which the a_i are the coefficients must be irreducible. That is omitted in the paper, but is typically understood. For otherwise, the field extension doesn't work, as you've found out.
When the a_i denote an irreducible, then the roots x_i are all outside Q.
And then there is no contradiction.
Your example uses (x-5)(x-3) which has all roots in Q--the diametric opposite--which is why sigma breaks down.
Digression: If you know some Haskell, you'll notice that a permutation on the roots basically fmaps to a (field) endomorphism on Q(all x_i). But here the converse is also true (exceptional in Haskell, except for trivial cases): every such endo comes from a permutation. (end of digression)
The second abuse is in the title. This is really "(My Opinion on) How to Teach Galois Theory to Undergrads" with a subtitle of "By Jettisoning the Fundamental Theorem and Focusing Exclusively on Quintic Unsolvability." The subtitle is omitted and the title shortened and de-colloquialized to read "Galois Theory for Beginners." This is all part of the prestigious mathematical tradition because ink, paper, and papyrus once upon a time were terribly scarce. Sorry about that.
Quintic Unsolvability is like FLT. The big prize is not the Yes/No answer but the VIP theorems--the statements of which are neither as easy to explain nor understand as QU nor FLT--used to nail down some pesky boolean.
So throwing out the FT of GT shortchanges the undergrad. It especially shortchanges the math-aware software professional who would appreciate experiencing the galois correspondence which later morphs into an adjunction in category theory. Quite cool.
GT has pedagogical messiness like inseparable extensions which can be skipped on a first pass. As a royal road to FTGT, I recommend the approach of fixing all fields as subfields of the complex numbers. See Postnikov's Foundations of Galois Theory available on google books the last time I checked. Nice exercises too.
p.s. (Galois) adjunctions are like a general theory of "How to Run Anything Backwards Even When There's No Chance in Hell." That's the power of math for you.