That's interesting! I thought only the queen was left from the past to play ceremonial roles. Do British people argue about the necessity of the post of the Lord Mayor the way they do it about the queen and the monarchy?
It's not something that is much debated. Most British people enjoy seeing our traditions being carried on, and they certainly contribute towards helping our tourism industry. Having said earlier that The Lord Mayor's roles were largely ceremonial and representative, I would emphasise the importance of the latter role. Like it or not (and many people in this country appear to disapprove) the financial institutions in the City of London make this country a great deal of money. Lord Mayors generally come from that background, are elected by their peers, and spend a great deal of their year in office promoting our financial industry around the world, often as part of government-sponsored delegations. The expenses of the office of Lord Mayor are largely paid for by these same City financial institutions - who would soon complain if they thought that they were not getting value for their money.
The official duties of Lord-Lieutenants (the Queen's representatives in each of Britain's Counties) and High Sheriffs (modern descendants of the mediaeval ‘Shire Reeves’ who were responsible to the king for the maintenance of law and order within the shire, or county, and for the collection and return of taxes due to the Crown) are very largely ceremonial these days. Neither are elected posts, but the holders are appointed by the monarch on the recommendation of county committees, and much of their time is spent supporting local charities and other good causes. Neither Lord-Lieutenants nor High Sheriffs receive any pay for what can be very busy schedules (up to 300 engagements a year for a High Sheriff, according to a recent government report). Lord-Lieutenants do get their travel expenses repaid; High Sheriffs get absolutely no expenses paid whatsoever - their cost to the public purse is zero.