Virtually every state secretary of the state in this country exercises similar statutory authority when certifying people eligible for ballot access and when they certify election results. All this AZ statute does, is give some specific guidance to its SOS with respect to this issue. So, Brewer is either uninformed, or is being intentionally deceitful - the SOS already wields this power.
Moreover, there is PLENTY of established case law where a state secretary of state disallowed ballot access on the basis of a constitutional infirmity to hold (or run) for a particular national office. There is an appeals process - it's called litigation. If someone is disallowed from from the ballot, they can sue the SOS in civil court. It happens, almost every election cycle. Again, Brewer is being wholly disingenuous.
Your points agree with what my perception of things were. I just wondered if there was any "special" enumerated difference in the bill that might make it different from "standard operating procedure".
It looks like someone or some group got to her.