I have this terrifying vision of a version where it is biases -- not diseases -- which spread throughout the world, and whenever a player's piece is in a city infected with a certain bias, they have to play with it..

Possible Arkham Horror variant: each increase of the Terror Track or Doom Counters infects a player with a bias. Closing a gate gives each player a single opportunity to guess the bias of a freely chosen other player, with a correct guess removing the bias. Sealing a gate additionally allows for the removal of a single player’s bias, even if nobody guesses it right.

Alternatively, just let everyone make a single guess in the Movement Phase, as per the normal Biased Pandemic rules.