This thread seems to be confusing typically-developing children who are not yet potty-trained, with parents of a special needs child dealing with toileting challenges. There is a big difference. I think the term "potty-trained" is what derails this conversation. I agree there are many pre-schoolers who are not fully potty-trained, and rules are in place for that reason.
However, when directly related to a disability, the more appropriate term is "incontinence." And yes, if the incontinence is a direct result of a disability, then it is appropriate to offer a reasonable accommodation. And yes, because it is disability-related, the ADA does apply -- and it applies to child care settings whether formally licensed as a "daycare" or not. If incontinence is the ONLY barrier to a child's participation in the program, as long as the accommodation does not interfere with the regular safe operation of the facility, and does not require staff to perform duties not available to the non-disabled, the accommodation must be considered.
A properly-fitting pull-up on a child (or adult) with continence issues will not be any more noticeable or impactful to others than a typical pair of underpants on those of us who get ourselves to the restroom and care for our own toileting needs. Use of a pull-up along with a requirement for parents to remain onsite and return at pre-determined intervals, results in no impact to staff duties and responsibilities or the safe operation of the facility and safety of other participants. It is my understanding that while proof cannot be required to determine disability, proof of the parents' responsibility towards the accommodation can be required (for example, proof of an ADR and cell phone number where they can be reached).
When mentioning "potty-trained" people seem to immediately think of dripping wet accidents and stinky messy diaper blow-outs. A special needs child with continence issues, who is wearing a properly fitting pull-up that is changed at appropriate intervals, is no more likely to present any of those health-hazard issues than a fully potty-trained child. I cannot fathom a parent of a special needs child with continence issues who would intentionally allow that child to sit around in a soiled pull-up for hours while they disappear for an evening at a theme park. Such parents are well aware of how frequently their child requires a pull-up change, as well as the unpleasant consequences of not changing it regularly, and are not likely to allow that to happen. Such parents likely would be more than happy to make their evening plans to stay within the resort for dinner and return to attend to the child's needs every couple of hours.
I have not used the various child care clubs at WDW resorts, though I do have experience with this issue in a general child care setting. The WDW programs are not exempt from discrimination laws. If they are truly excluding a child solely on a disability-related factor that can be reasonably accommodated, they are opening themselves up to a legal challenge. I'm going to guess that hasn't happened yet, it doesn't mean they are in the right to deny access or a reasonable accommodation.