A mayor's power is often seen, even when compared to a governor's or prime minister's/president's power, as having the highest potential of actually being appreciated, as the latter positions come with having a bunch of invisible pieces and filters to tend to, even supposing you decided to be dictatorial about things. Despite this, or maybe in spite of this, whenever I see very loved and communal individuals, they see it as above their area of motivation to run for local office. There isn't a single city, town, or village I've been to where the mayor's level of connection to the people around them isn't overshadowed by that of at least some of the citizens, in fact I see the mayor, district attorney, sheriff, town judge, etc. in my own area as being visibly condescending blowhards who are bedfellows with the local activists who are known to have no issue ruining childrens' lives the Ally Bank way. Even to you I'd recommend running for some form of town office, though with you too, I doubt the challenge would be stepped up to. You could make a difference in your own little fragment of the world.
So considering most people I talk to wouldn't take up the suggestion to run for something like mayor, district attorney, sheriff, town judge, etc. what is your local government scene like? And are you different from those who won't step up to the challenge?
The funny thing is he might be the one in the right, if we examine with logic.
A mayor probably has no particular skills or abilities to help personally in that sort of situation, and if he was doing his job correctly in the past, then everything in his power to do would already have been done. The appropriate experts are ready. Emergency plans are in place. There's backup plans, and backup plans for the backups, all carefully considered and planned by the best people the mayor could get to do them in the past.
But humans are weird and have stupid ideas, so we want to see the mayor in his office, giving interviews, or even better, at the site of the disaster, helping. Except the most the average mayor is likely to do in an emergency situation is get in the way and be a distraction.
That's the sad thing - your city's mayor may or may not have done a good job, I have no idea, but the average mayor will definitely be attacked if this happens and he doesn't make a show of 'helping'.
The mayor's job is to co-ordinate public information and make press releases during an emergency. He's an authoritative source. Better for people to get info from him than from rumours and hearsay.