The purpose of the sump pump isn't to pump water out of the basement
if it fills up with water. It's purpose is to collect water from the
drain tile and pump it out of the house. If you took the cover off,
you would see the drains running into the sump crock.

I understand that the "Main" purpose of a sump pump is to pump out
water that is never actually in the house to begin with, but isn't a
secondary (and potentially very useful) usage to pump out water that
does somehow accumulate in the basement? I know that growing up,
whenever we had water heater issues, and the basement flooded, the
first thing we did was to squeegee the water over to the sump pit so
the pump could do it's thing.

You should manually run the sump pump for a few seconds every few
months so that the bearings do not seize up...

This is what I want to do - to test the sump pump to make sure it
works, regardless of whether it's to pump out basement water, or water
from the drain tile. My question is: Is it worth pulling up the caulk
to test it, only to have to re-caulk it afterward? Can I bore a hole
in the top of the sump pit cover, test it, and then plug that hole
with an airtight (but not caulked) rubber plug?

Thanks again!