You're generally correct, except that deed polls aren't used in the US - they mainly exist in Commonwealth countries. But yes, the US will be fine with her using any name that reasonably describes her and that she uses habitually.
For passport purposes, the US does not require a separate legal name change procedure for applicants who get married or divorced. The marriage or divorce certificate on its own can be used as proof, and an applicant can use either the current or previous name as she wishes, if that's the one she uses in everyday life.
Her passport is not fraudulent at all. If she's American, the US State Dept has always accepted that people may use different names at different points in their lives, and specifically permits divorced citizens to continue using a married name in their passport after divorce if that is a name that they've habitually used for at least five years for other purposes.
You probably looked at something like this and overlooked the part that says "once you have your letter...". The "visa on arrival" system that they're talking about (which has basically been defunct since covid) was only for people with pre-approval letters to get a visa affixed to their passport at one of those airports - it NEVER allowed Americans to just show up without prior authorization and get a visa upon landing.
I've gotten about half a dozen e-visas from the official site and that has never happened to me - it always prints exactly what I typed on the application.