First, let's examine definitions under regulatory law, specifically 28CFR36.104:
"Disability means, with respect to an individual, a physical or mental impairment that substantially limits one or more of the major life activities of such individual; a record of such an impairment; or being regarded as having such an impairment."
"mental impairment means -- ... Any mental or psychological disorder such as mental retardation, organic brain syndrome, emotional or mental illness, and specific learning disabilities...[including but not limited to]...mental retardation, emotional illness, [and] specific learning disabilities...."
"Service animal means any guide dog, signal dog, or other animal individually trained to do work or perform tasks for the benefit of an individual with a disability, including, but not limited to, guiding individuals with impaired vision, alerting individuals with impaired hearing to intruders or sounds, providing minimal protection or rescue work, pulling a wheelchair, or fetching dropped items."
Note that the definition of "service animal" does not in any way restrict the meaning of "disability" with a qualifier such as "physical disability" or "some disabilities." Therefore, since both terms are defined in the same section of the same act, it is not reasonable to apply only part of the definition of "disability" and not all of the definition of "disability" in the meaning of the term "service animal."
It is important to make a distinction between ESAs (emotional support animals) and PSAs (psychiatric service animals), because PSAs are task-trained just like other SAs and are not there solely to provide emotional support. Therefore, PSAs meet the requirements of the definition of "service animal." People often confuse ESAs with PSAs, which has probably lead to this misunderstanding of whether the use of PSAs/PSDs is covered under the ADA.