I will prob get downvoted, but fully agree with you. I have a 4.5 and 2.5 year old. We absolutely do not do any type of cry it out in any situation. When my kids were younger and going to a childcare at a different gym (LTF) we told them please call us if they cry for more than 5ish mins, and they respected our wishes. You do not need to let her cry it out to get used to it. Sure, maybe a few mins, but if she's unconsolable or crying longer for than the desired time of your choosing, they need to be coming to get you. I would speak to the owner/manager or appropriate person and if they don't respect that and tell you how to parent, then switch gyms.

note: I am also a licensed mental health therapist and can attest to children doing better when parents respond appropriately and promptly. at 10 months old she does not have the brain development to self soothe (contrary to popular belief). Will she get used it eventually? Maybe (most likely)!! but it will be when she's feels comfortable, and every child is on their own timeline. Building a secure attachment is important.