Doctor's appointment

Alice, ‘ Il fait des visites à domicile ‘ mais ‘ il rend visite à qqn ( or even ‘ il rend visite à qqn à leur domicile ‘ ). 

It is the people the doctor is visiting, not the place.

When talking about visiting people ( ses patients ), ‘ visiter ‘ alone is not correct.  

The relevant section from the lesson :  

Note that the verb visiter is used with places.  

When visiting people (friends, family...), we use the expression rendre visite à [quelqu'un] and never visiter alone. 
(This is similar to the English to pay a visit to [someone].)