If we got rid of the word Lord to refer to human beings pretty soon we would need another word to translate the Greek Kyrie and the Latin Dominus.
Of course, if this mayor is a “she” it’s absurd to call her Lord Mayor rather than Lady Mayor, but that’s another subject.
Certainly there is a danger of pride with these sorts of titles. That’s part of why we Americans got ride of them (the association with the condemned idea of inheritance of titles was another reason). Then again there is something to be said for vesting important positions of authority in a kind of majesty, so long as this remains firmly rooted in a republican spirit of government. Even in our country we chose classic titles like Senator or British Parliamentary terms like Speaker for some roles, and the office of President has evolved its own proper dignity.
However, you Brits have kept alive many of the old Medieval terms, and I don’t think this is the right hour to do away with them. Our culture has become so forgetful and dismissive of the past that reminders of continuity, even if they are silly things like titles of nobility and related terms like “Lord Mayor,” are very valuable. As I’ve mentioned it also keeps these old terms alive for the entire English-speaking world, avoiding the awkward problem of having to consider whether it is time to start referring to “Governor God” instead of “Lord God” or “Christ the President” instead of “Christ the King.”