I am right in picking my battles, but you should probably read the entire part thing:

(6) Inquiries. A public accommodation shall not ask about the nature or extent of a person's disability, but may make two inquiries to determine whether an animal qualifies as a service animal. A public accommodation may ask if the animal is required because of a disability and what work or task the animal has been trained to perform. A public accommodation shall not require documentation, such as proof that the animal has been certified, trained, or licensed as a service animal. Generally, a public accommodation may not make these inquiries about a service animal when it is readily apparent that an animal is trained to do work or perform tasks for an individual with a disability (e.g., the dog is observed guiding an individual who is blind or has low vision, pulling a person's wheelchair, or providing assistance with stability or balance to an individual with an observable mobility disability).

This means that if it's apparent as to WHY the service dog is needed ... the guest is obviously blind. The guest is obviously restrained to a wheelchair - then you're right - paperwork or verification can not be asked. But aside from obvious disabilities, it IS PERMITTED to ask for paperwork. So no, I am not wrong in asking for the badge work, especially in the private sector.

Just to make sure we're crystal clear here, the loophole is stated in the literature you provided:

Generally, a public accommodation may not make these inquiries about a service animal when it is readily apparent that an animal is trained to do work or perform tasks for an individual with a disability (e.g., the dog is observed guiding an individual who is blind or has low vision, pulling a person's wheelchair, or providing assistance with stability or balance to an individual with an observable mobility disability).