They have an public obligation to provide a service, even if a limited service!
They provide a scheduled service, if they don't schedule for Christmas day then they are not required to.
Last time I looked REX was not a "private entity"... they are a listed public company. Their stakeholder obligations do not stop at their shareholders
Yes they do. If not enough people want to fly on Christmas day that REX can't make a profit then it would be irresponsible for them to operate.