The style of singing you describe is known as "intoning".
Here is a description from online copy of
["A Dictionary of Music and Musicians (1900) edited by George Grove", article by William Smyth Rockstro]
INTONING. The practice of singing the opening phrase of a Psalm,
Canticle, or other piece of Ecclesiastical Music, not in full chorus,
but, as a solo, or semi-chorus, assigned either to a single Priest, or
to one, two, or four leading Choristers. The term is sometimes
strangely misapplied. For instance, we are constantly told that the
Litany, or even a whole Service, was 'intoned' by some particular
person; when the word used should have been, in the one case, 'sung,'
and, in the other, 'monotoned.'
You mention "vicar", so I assume Anglican (Protestant) context here, but the practice is common to other Christian traditions, since it originated in Gregorian Chant.
Here is a historical observation on the (probable) evolution of the practice of intoning in the Anglican tradition : because the congregation were expected to join in all the service, but were not familiar with the chants, the "parish clerk" would lead the responses ["The Services at the Communion Table Considered" by 'Philobiblion' pub London 1842 ] (https://books.google.co.uk/books?id=Li5hAAAAcAAJ&pg=PA77&lpg=PA77&dq=parish+clerk+intoning&source=bl&ots=CaKAEaMjBH&sig=ACfU3U0Kfhi_B1Eq0cDy31Idb5D5Spkr_w&hl=en&sa=X&ved=2ahUKEwiyl4ynypniAhUsShUIHQTTBN8Q6AEwDHoECAgQAQ#v=onepage&q=parish%20clerk%20intoning&f=false)
You mention the harmonic "oddness" of the melodic line ("often in a minor key and perhaps even atonal melodies.").
This reflects the plainchant origins of the style. Plainchant was modal in character. Modes are like the scales we are more familiar with inthe Western tradition, but they start and end on unfamilar notes. So, they sound odd to modern ears more familiar with scales, but this oddness is used by more modern musicians for effect, for example jazz musician Miles Davis.
From the same musical dictionary above, the article on
gives examples of specific "intonations" for particular psalms or other parts of prescribed pieces of music for certain religious services. The article also mentions the use by composers such as Handel ("the Lord gave the Word" from "The Messiah") and Mendelssohn ("Lobgesang"), but many settings of the Mass or works on religious themes by classical composers will have fragments of intonation present.
examples: some overlap with Plainchant, inevitably.
This is a good worked example of how the text is sung to a limited set of notes.
Other traditions :
Personally, I can recall as a child in the 1960s, the practice of intoning being common, but nowadays, I think it is only to be found in churches which adopt a "traditional" form of worship, or where it is revived as part of respect for tradition.
I hope this answer will allow you to investigate, and make your own discoveries