Yes, we usually use the indefinite article when we describe someone's occupation in a construction with be:
He is a doctor.
But when we name a unique role or task, we either use the or omit the article altogether:
John is (the) captain of the team.
Since the team has only one captain, we are free to omit the.
When we put the person (he, John) and the "describing element" side by side (apposition), the same principle applies:
John, (the) captain of the team, said: "We are going to win this match".
Mr. Smith, a doctor, works...
... is okay.
But since "public health nurse" is not a unique role or task, we should use a:
Ms. Smith, a public health nurse, was born..
"Chief Justice of the Supreme Court" is a unique role, so the is omissible:
Mr. Smith, (the) Chief Justice of the Supreme Court, did..
But we do not drop the if it stands before an ordinal number that describes a person's position among other persons of the same rank, hence
Mr. Smith, the third Chief Justice of the Supreme Court, did..
P.S. If you're interested, in Unit 5.37 Quirk et al. start off by describing noun phrases in copular relationship (constructions of the type "Mr. Smith is a carpenter", where 'is' is a copular verb). Then they point out that you can omit the indefinite article in the following "as-constructions":
her duties as (a) hostess
my appointment as (a) lecturer
Jung as (a) thinker
It's interesting why exactly we can omit a in these cases. The authors write "compare with 'unique role', Unit 5.42". Could "hostess, lecturer, thinker" be understood as "unique roles" in specific contexts? I don't know. Article usage is hellishly complex.