There is a category error here.
The fallacy is the idea that a member of the public can stand for "the public" and thus command service from a public servant.
That is, a member of a set is not the set and does not inherit properties of the set. A member of a set, and the set, are different categories.
To borrow a motivational example from computer program design: If B is a type of A, then it is not true to say that collection-of-B is a type of collection-of-A. So, even if B and C are both types-of A, and you can substitute a C for a B, you can't substitute a collection-of-B for a collection-of-C. And you need to watch out for the error of trying to park a nuclear powered aircraft carrier in the spot reserved for electric scooters. Even though you could use either the aircraft carrier or the scooter if all you needed was "a vehicle."
"The public" is a set. In general, it is an abstraction referring to a group of people with shifting membership. As people are born, achieve age of majority, move to or from different countries, and eventually die, the group is changing over time. It is the idea of the people living in a polity. It does not refer to any specific individual.
If the term "the public" does not refer to something very close to this, then the idea of "public servant" does not make sense in the situation. For example, in a totalitarian dictatorship, a "public servant" would not be a sensible title.
A "public servant" is, in theory at least, a servant to the abstraction "the public." For example, in some polities, such persons swear an oath "to the constitution" or "to the crown" or to some other symbol of the community or country.
A "public servant" is thus not a servant to any individual in "the public" since a member of a set is not the same as the set. A member of a set is not of the same category as the set.
Indeed, in political matters, the set of "the public" may easily be in conflict with a single individual. (Whether that conflict is moral or ethical or desirable could lead to a huge collection of interesting philosophical discusssions.) This is a significant portion of the reason that there are such things as public servants, in order to enforce the decisions of "the public" (arrived at by whatever means currently in exsistence, again leading to huge philosophical discussions).
A police officer is a primary example. Through some means, laws have been enacted. (Keep those swaths of philosophical disucssion in mind. The laws may or may not be moral, ethical, desirable, etc.) These laws are presumed to be the will of "the public." Being a "public servant" means the police officer must follow those laws, not what any one member of the public says.
The police officer is a public servant.
Abbot is a member of the public.
The police officer is not a servant to Abbot, because Abbot
is a member of the public, but he is not "the public."