I don't know if that's really correct. In your sentence, there is an implied "office" (doctor's office) because you're making doctor possessive. But medicus is the person, not the location, and there is nothing in the Latin sentence to imply possession.
Your sentence would be something along the lines of: Femina aegra ad hoc medici it. (to this [place] of the doctor).
I think it's just idiomatic. Of course, the doctor's office is implied, but it's the same meaning, as when you go to see a doctor, it's in the doctor's office.
In English, when you use "someone's " with no word behind, it's not really a possessive anymore. If you mean the meaning of the French word "chez", it means the same.
You link re-affirms my statement... The posts in that forum say "at the doctorʻs" means "the doctorʻs office". What I mean is that "doctorʻs" refers to the building and not a person. A suitable substitute for "doctorʻs" would be "hospital" or "clinic". The latin sentence is specifically referring to the person (doctor), not the location. Imagine this was a student at school, they might say they are "going to the doctor", meaning the school medical person. If, instead, they say "going to the doctorʻs", everyone would assume it was someone at a different location (such as a hospital).
And to furthur clarify my intent, saying "someoneʻs" with no word behind it almost always means a location and not the people themselves. If I say, "Iʻm going to Juliaʻs," that says nothing about whether or not Iʻm going to see Julia. It only means I am going to her house/home.
You can say "The woman is ill" or "The woman is sick" (they are not perfectly the same), as an attribute, using the verb, linked with the verb to the subject.
But to use without a verb, before the noun, as a non-attribut adjective, you won't use "ill" but rather "sick".
an ill child
a sick child