I am not saying it is not applicable. Whether the FCA approach is sutiable in your case, depends on several factors, such as
(1) what is your (research) question/problem you would like to answer? If you just want to know a service area of a hospital? Or do you want to match demand for hospitals/doctors with supply? etc.
(2) you need to take into consideration the national health regulations. For example, many countries defined performance indicators like "maximum travel time/distance" to next hospital. These parameters should be accounted for in your analyses. And yes, in some countries patients don´t have a free choice to select the general practitioner or dentist or even the hospital (in other words, the match/assignment between supply and demand has already been decided for by state-government when the government already defined some kind of 'health care zones' - sometimes the definition of such zones already accounted for accessibility, but often such zones where defined based upon general administrative units or other factors).
(3) you may also differentiate between emergency situations (i.e. a patient is transported by rescue services; in this case usually patients are not be asked where to go) and 'normal' situations where patients drive themselves to hospitals/doctors.
(4) you may even also look into the type of medical treatment. Usually, not all health facilities are capable for all kind of treatments, so specialized treatments may only be offered in selected facilities. In such cases, a purely drive time area approach may also not adequately reflect the problem statement.
(5) as part of their health policies, many countries/regions defined also indicators like "number of hospital beds per inhabitant", which at a glance seem to be an easy indicator. But as the actual corona pandemic shows, you also would need to take into a account a temporal dimension (number of available hospital beds per inhabitant at a given time - this usually is often not accounted for in the regulations).
You should think about these (and probably many more) research questions and answer them before you assess whether the FCA approach is suitable or not (and probably how it needs to be modified to best match your research question), and how and where the network analyst can be used to answer your questions.