Indeed a terrier bag is an accessory for a ‘short legged’ terrier. Long legged terriers run with the pack. (The whole ‘Parson’ or just plain ‘Russell’ terrier thing is an AKC abomination, real foxhunters just call 'em all Jack Russell terrorists).
Actually what they are often meant to do is bring a fox to bay underground, the noise is used to locate the fox so the terrier men know where to dig in order to remove and dispatch the fox (I speak of UK here where yes, fox were and are vermin and if not dispatched this way, farmers just shoot or poison, or at least that was the scenario pre-ban. These days I perceive the foxes just move to town and feast on small children).
Back in the day, a terrier ran with the Bull Run hounds, and indeed she would go down the hold and bay at the fox. Of course we weren’t digging out and could never get her to come out timely, so we’d continue to next cover and she’d catch up 30 minutes or so later, I guess when she finally figured everyone had gone and left her!
Somewhere I have a picture of a JRT hunting with the Red Rock hounds- he just perched on the horse’s back behind the rider and ‘surfed.’