Should gynos be doing mouth checks since oral cancer from stds is on the rise?

Yes: But not for that reason. They are considered a primary care specialty so should do a complete physical exam which includes the mouth. Some women don't see any other physician.

While I am a: Cancer physician, and i know what diagnosis my patients have, i tend to do a physical with a focus based on what the patient tells me. Hpv associated cancer in the head and neck is associated with the oropharynx: tonsils, tongue base, pharyngeal walls, suprglottic larynx...You cannot see most of this without a mirroe or a scope. Oral (tongue, floor of mouth, buccal mucosa, etc) best by dds/ent.