Yeah, I think we've covered this all before. The problem with polearm master including quarterstaff is, at it's root, a problem with the weapons table.
Club - Light - d4
Mace - nil - d6
Quarterstaff - Versatile (d8) - d6
Greatclub - Two-handed - d8
The problem is that "quarterstaff" is portraying two weapons: dueling cane and quarterstaff. But you can't do that because the weapons table isn't granular enough to put cane on the table and be mechanically distinct (setting aside how many indistinct weapons are on the list, especially at martial level). Essentially, the game is making quarterstaff be a normal quarterstaff, and a bo staff, and a jo staff, and a dueling cane, and a baseball bat, etc. The club is basically the size of a belaying pin or truncheon only. A mace is any shod or metal club. Anything long enough to be versatile is automatically a quarterstaff. Of course, mechanically, quarterstaff already invalidates both mace and greatclub, which really is stupid, but lots of people have problems with the 5e weapons table (e.g., why is "dart" still on the table instead of replacing it with throwing knife? Nobody knows what war darts are except war gamers!).
Breaking suspension of disbelief doesn't require doing extraordinary things. It requires breaking the conceits of the narrative. Here, one of the narratives is that melee combat is generally confined to the forms and styles of combat consistent with historical Western or European martial arts prior to the 16th century as portrayed in the mass media of the late 19th century through to the modern day.
Yes, you could in theory use a shield and then use a quarterstaff like a spear with no blade. The historical reality is that nobody carried a shield unless they were going to war. That means they were carrying a weapon built for war, such as a spear. A staff -- quarterstaff or otherwise -- is a weapon of convenience and social acceptability, not a weapon of war. That's why staves are weapons of monks, peasants, woodsmen, and gentlemen defending themselves from brigands. Nobody ever fielded regiments of quarterstavers on purpose, with or without shields. So it breaks suspension of disbelief.
