Im wondering whether it's to do with health and safety - my gym is Virgin Active in Sheffield which is genuinely the most family-friendly, breastfeeding-friendly place I've been to, but they do ask you to move to the poolside if you're feeding. I'm sure they wouldn't do that if there wasn't a good reason, since the whole place is full of breastfeeding mothers (it's the only place with a crèche for miles).