Alan Gerber wrote in
:

I assume the purpose it serves is it tells them when they can
terminate flight following.

Alan,

VFR flight following is workload permitted basis.

So, like you, I have experienced the same, approach asking me to advise
airport in sight. Approach can tell you to squawk VFR, have a good day
well before you have a visual on VFR flight following. In my area, I have
never had that done except once when I was arriving after the ATC shut down
for the night.

Generally IN MY AREA, if it's that busy, approach much rather you squawk
and talk then not. But I am only dealing with C airspace.

For IFR, a visual approach requires sight of the airport / runway
environment, and ATC cannot terminate your IFR until you request it (I.E
cancel IFR like in my case at an uncontrolled airport) or land at a
controlled airport.

Allen