I agree with both of you, but I think I'm somewhere in the middle. I agree that the Burgomaster is underwritten (based on the summary that was posted), and that running him perhaps requires a bit too much from the DM, leaving the door wide open for dreadful mistakes. However, I also think that it is fine for a campaign to have unswayable pure blackhat npc's. Not every npc needs to be complex. But I wonder what the point of the Burgomaster encounter is (I haven't read or played this particular adventure). It sounds almost like he is purely there to force a conflict on the players.
Like Iserith, I use a slider for my npc's during social encounters, although the slider is not literally visible. However, I do communicate to my players whenever an npc is swayed by their arguments, or when they anger the npc. It is important for the players to know when they are making progress, or losing progress during a social encounter. I also point out nuances during social encounters, such as an npc being disinterested, highly negative or very passionate about a topic. It is important that the players get some clues regarding what an npc cares about, so they can make an informed choice regarding what they say to the npc.
Some npc's may already start out with a negative view of the players, making them much harder to sway. While others may already be friendly to the players, making them far easier to negotiate with. And there may be some topics that a particular npc is impossible to sway on, which I always point out to the players during the encounter so they can move on. I also keep track of so called loyalty quests for various factions (an idea I stole straight out of Mass Effect), to progress the attitude of a group of npc's further to the positive. The opposite is also possible, where certain quest outcomes will permanently make some factions hostile to them.