It seems as though the pillagers were spawning too far away from the village or villager or the player, and were unable to pathfind to the village.  When the bad omen effect was dropped by drinking milk, the pillagers never despawned, eventually filling the mob cap.  

I attempted to draw the pillagers in to a trap to kill them, but this would result in the bad omen effect returning and more pillagers filling the mob cap.

When the pillagers filled the mob cap, no other hostile mobs were spawning even though I was playing on hard difficulty.  

Only after killing the pillagers with a /kill command did the hostile mob spawning eventually return to normal rates.