Periodic reminder that 'the game is afoot' does not mean 'the game has begun'. It means 'the prey animal is running'. Game is used here in the sense of gamekeeper, game pie, game bird.

The full Shakespeare quote, which Holmes only uses the last bit of, refers to coursing with greyhounds. So the 'game' was probably a hare.

See also Henry IV: 'Before the game is afoot thou still let'st slip.' In other words, 'you're letting your dogs off their leashes before the prey has even started to run.'

this isn't meant to be 'you're WRONG', it's 'hey there's hidden context and history here that you might be interested to know'