They do when the person is trying to get underneath the ADA for protection or benefit, i.e. getting an emotional support animal or a handicapped parking space. If they are being treated for virtually anything chronic, it falls under the ADA definition of disability. In this case, they are trying to play in a sports league. I don't see how the situation wouldn't affect one or more major life functions to satisfy the definition.
I don't see why they wouldn't want to go this route to meet their objective, other than the stigma of being disabled. But virtually no disabled community enjoys being labeled as disabled, they view themselves and their community as whole and normal (deaf, blind, physically handicapped, etc.). But the law views them as not functioning normally for the specified reason, for their own benefit and protection. Here, get under the ADA and boom, reasonable accommodation must be granted and they are allowed to play.