I have used the baby care centers many times with my nursing daughter at 3,4,and 9 months (and 6 months in DL). At 3 months, my cousin and her nursing 6 month old were with us, and they both ate every 2 hours so since we got up the same time and stuff, they ended up getting hungry the same times, so we'd head off to the baby care centers together while the others did a ride. That was May though and it was very warm so I was glad to get out of the heat, plus have a nice place to change her diaper and let her have some tummy time on the changing table since they're so big.
At 4, 6, and 9 months it was just me and the baby (and my husband!) so I nursed wherever on those trips. Still used the baby care centers if we were near one, but if not then we nursed on a ride/show or a restaurant. Never really just used a bench cause we were always 'somewhere' when she got hungry.
We're going back in April (18-26) and Madison will be 13 months and is still nursing, but mostly daytime self-weaned. She usually nurses at naptime, bedtime, and then during the night, so we'll probably just nurse in the room. At this point she will usually nurse if she's hungry/sleepy, but at 6 and 9 months it was horrible with the distractions. Even the baby care centers were bad cause she'd want to see the other babies!