In connection with the use of article in the NT, BDAG says: “In its individualizing use it focuses attention on a single thing or single concept, as already known or otherwise more definitely limited: things and pers. that are unique in kind.”
In our case the following would apply: “In its generic use it singles out an individual who is typical of a class, rather than the class itself. So also in parables and allegories. The generic art. in Gk. is often rendered in Engl. by the indef. art. or omitted entirely.” So the GB would become a class of its own. They take on a huge responsibility.
Interestingly, they do not comment on the hypothetical third class condition of the follow-up verses. In both Matt 24:45-51 and Luke 12:42-46, the eschatological disqualification of the servant is a distinct possibility because of misconduct, with dire consequences for the servant.