"Experienced doctors and nurses helped them."
Translation:Spertaj kuracistoj kaj flegistoj helpis ilin.
It is only an academic degree, much like it originally was also in English. So a physician (sanigisto, kuracisto) can be a doctor (doktoro), but may as well be a magister (magistro) or a professor (profesoro) of medicine. On the other hand a doctor (doktoro) can be a physician (sanigisto, kuracisto), but may as well be a mathematician (matematikisto), a lawyer (juristo) or a linguist (lingvosciencisto).
What about in an academic setting if they have a doctorate but not a position as a professor?
I'm a mathematician, not a lawyer, but in my case, since I teach at a community college (and this is my only academic position), I'm not properly called ‘Professor’ (although students do call me that sometimes). Still, I have a doctorate, so in that context, I am properly called ‘Doctor’. (This is in the USA.)
I agree that only physicians (broadly speaking, including dentists, osteopaths, etc) are called ‘Doctor’ outside of an academic context, at least in the USA.
I'm also in the USA, and my experience has been that students call any college/university instructor "Professor So-and-So" by default, unless explicitly instructed otherwise. The "otherwise" is usually to address them as "Doctor So-and-So" in recognition of their degree.
A notable exception to this was the Honors Program I was part of. That department collectively decided to cultivate a very relaxed, informal atmosphere and insisted that the Honors students all address them by their first names.
I suppose you wanted to ask about phlebotomist; there's no such a word as phlebotanist. :D
Esperanto word fleg·ist·o is based on the verb fleg·i (to nurse, to tend), which comes from the German verb pflegen, which means the same thing.
English phebotomist is a profession name from the noun phlebotomy, which comes from Greek φλεβο- (phlebo-, venous) and τομή (tomḗ, incision). You could use it in Esperanto on the basis of the rule 15 of the Fundamento, so have flebotomi·o and flebotomi·ist·o (don't mix it with flebotom·o, which is a sand fly, and flebotom·ist·o, which would be a name of a scientist, who studies sand flies). However, those words would be understandable only for people who know this medieval pseudo-treatment and in their native language it is called similarly (I didn't know what is a phlebotomy, even though in Polish we apparently have a word flebotomia). So to be well understood, you could also use currently existing words in Esperanto and call it for example sang·el·tir·o and sang·el·tir·ist·o.
Now, when I know what phlebotomy is, I think that you've maybe wanted to ask not about phlebotomy, but just ordinary venipuncture (German Blutentnahme, French prélèvement sanguin). :D In Esperanto venipuncturist (a nurse performing venipuncture) could probably be sang·kolekt·ist·o, but I didn't investigate whether there already is any well-established word in use.