As Professor Joad would have said "It depends what you mean by Rabbi".
Wikipeida uses the traditional meaning of Rabbi as:
a teacher of Torah. This title derives from the Hebrew
word רַבִּי rabi, meaning "My Master", which is the way a student
would address a master of Torah. The word "master" רב rav literally
means "great one".
But the article goes on to say:
In more recent centuries, the duties of the rabbi became increasingly
influenced by the duties of the Protestant Christian minister, hence
the title "pulpit rabbis", and in 19th-century Germany and the United
States rabbinic activities including sermons, pastoral counseling, and
representing the community to the outside, all increased in
importance.
and it is this “minister” that you are referring to when you say “congregational (synagogue) rabbis”.
The answer is, as hinted by Shokhet's comment, that a Rabbi is not needed for a funeral etc. A reasonably able layman can perform the job at least equally well.
{Pet peeve – think of the “Rabbi” who asks for notes on the life of the departed so he can make an eulogy when the one writing the notes could have spoken more personally.}