It is common in journalism to start a sentence with all or part of the thing someone said, and indicate who said it later in the sentence. If Bonnie Glaser had already been introduced, or was otherwise familiar, the author might have written
As a Chinese-speaking democracy, points out Bonnie Glaser, Taiwan has long been valued by the United States as an alternative political model to the Communist-run mainland.
Journalists do this because the thing being said in these cases is more important than who said it. So they often bury the speaker's name in the middle of the sentence, which is the least salient location.
But this kind of sentence-rearrangement becomes harder to follow when the pieces get too big. That's what went wrong in the original sentence.
Added later: I should say something about when the rules of grammar actually allow this kind of rearranging. Unfortunately, this is one of those things where native speakers know what they can do without being sure how to put the rules into words. Of course, normally the subject (Bonnie Glaser) comes first in the sentence, followed by the verb. But verbs of saying or thinking have a lot more flexibility. All three of the following are acceptable. (And in these examples I used direct quotation, but it also holds for indirect quotation as in the OP's example.)
Jim said, "Ouch."
"Ouch," said Jim.
"Ouch," Jim said.
Note that, if your original sentence were "Jim ate a sandwich" then you absolutely cannot reverse it to "A sandwich, ate Jim."