I'll speak to the landscape of medical training for women first, and then about the specifics of how patients would respond to a woman physician. I am going to focus on the era that predates your grandma's training and practice (late 1800s-early 1900s), because I think this era says a lot about the beginnings of a culture of women in medicine that would have set the expectations of her patients in later decades.

In the late 19th and early 20th centuries, there were some medical colleges for women founded in the USA and the UK. Women were not typically admitted to medical schools which were exclusive to men. Even if the medical school did not openly exclude women, it excluded them by default because applicants were often required to hold an undergraduate degree before entering medical school, and undergraduate universities by and large turned away women. There are exceptions to this, such as Elizabeth Blackwell MD. She is widely considered the first female physician in the USA, who was accepted to Geneva Medical College in NY after her admission was put to popular vote amongst the all-male student body. Mary Edwards Walker (the first credited female surgeon in the USA) attended Syracuse Medical College; I am not sure of the circumstances of her admission, although as I understand it she received an excellent education in her earlier years which was unusual for women at the time, enabling her to master the scientific fundamentals needed for medical training in the 1800s. Dr. Walker applied to serve as a surgeon for the Union during the Civil War and was denied, although she ended up volunteering her services and received a Medal of Honor after being captured by the Confederate army and imprisoned.

As for the perception of female physicians, you are right that male physicians throughout history have been in general more respected and seen as having greater medical authority. But there are important exceptions to this generalization. Pediatrics have long been considered a field of medicine especially appropriate for female physicians, who were historically seen as gentle, motherly, and able to effectively care for younger patients. Though obstetrics were dominated by male physicians from about the 18th to the 20th century, primary care/GP for women patients would have been done by a female physician in instances where the patient felt uncomfortable with a man examining her. On the opposite end, highly technical specialties of medicine like surgery have always been closed, if not hostile, to female practitioners. In fact, surgical specialties today continue to be populated largely by men-- I think something like 5% of all orthopedic surgeons in the USA are women.

In sum, I am sure your grandmother faced many difficulties entering the field of medicine than women medical students today don't have to tackle. That said, by the mid 20th century, there were long-established medical colleges for women and enough female physicians in practice that her patients would have precedents. They may have thought that being a physician conflicted with her responsibilities as a wife or mother, but not that a woman could not effectively perform the job of a physician, particularly as a GP/psychiatrist in a less technical specialty.