A "night court" is simply a court that operates outside the ordinary business hours for the convenience of people who have work during those hours. They aren't common and are mostly found in medium sized and large cities.
A "night court" can do whatever judges in the jurisdiction decide should be offered at that time based upon the needs of people who deal with the court system.
Small claims civil cases and protection order cases, for example, are also often handled in a night court format, and a variety of criminal law matters not requiring a jury trial could be handled in that format.
Most cities have a judge or magistrate assigned to take arraignments, not always at night (often the position is rotated among judges in the jurisdiction every now and then), which, as the question notes, "should serve to read the charges, set bail and potentially record a plea, but defer the rest of the process (discovery, pre-trials, depos, trial, and eventual sentencing) to later appearances."
An arraignment court would often accept guilty pleas, and sentence people when they plead guilty to infractions, petty offenses, and misdemeanors. It would also often dismiss charges when the prosecutor decides that the case has no merit and decides not to pursue it. It would rarely enter not guilty pleas on the merits, but the court in the television show might not be a pure arraignment court.
But, as my knowledge of television comedies is less expansive than my knowledge of the law, I can't tell you if this is "an accurate portrayal of a Night Court, or purely a device for fictional purposes."
I'm sure that the TV version has wittier dialog, more interesting lawyers and court personnel, and less boringly repetitive cases than real life, but I'd have to defer to someone who has ever watched more than the trailers of the show to comment further.
