Getting official copies of a lot of these documents can take time in a situation when there is an obvious time limit. If a single official or a handful of officials with their own agenda can gum up the works, then the bill might be used against a valid Republican candidacy as an invalid Democrat candidacy. I had to go through a similar rigamarole when I first applied for a passport. If my parents weren’t around to sign an affidavit, I’m not sure what I could have done, because in their case, the US government never gave them a naturalization certificate to fill out. Now if Brewer vetoed simply because it was a bad law instead of just ideology or McCain variety political calculations, then perhaps she needed to make more specific legal objections to the legislature. But short laws can be abused just like 100 page laws. Whether McCain has Svengali-like powers over Brewer is beside the point.
IMO your fears are not well founded.
That’s why there has to be a process to hold the “gatekeeper” legally accountable. This bill would have allowed any legal resident to challenge the SOS’ decision - effectively taking it out of the SOS’ control and putting it in the hands of the court, which is responsible to adhere to the law and to evidentiary standards.