transponder not working

A few days ago I was getting ready to do a cross country with one of
my students. This was going to be his first cross country, so the
focus was going to be on pilotage, dead reckoning, planning, cockpit
management, and stuff like that. I was going to save the flight
following, flight plan, class C stuff for a later lesson.

He planned to a class D airport about 25 miles northeast of SFO, away
from the Bravo, but still under the mode C ring. He seemed to have his
stuff together, so after a few checkpoints I decided to go ahead and
call up Norcal to get following. I thought "might as well".

So I show my student how to get the frequency out of the AF/D. Then I
give them a call, tell our location, altitude, type, etc. The
controller gives us a squwak code and wait for him to tell us he has
us on radar. A few seconds later he calls us back and tells us to
recycle our transponder, as he isn't seeing anything on the scope. I
recycle it a few times, check the circuit breaker, and anything else I
can think to do. He comes back saying he doesn't have us, and I just
respond, "thanks anyway".

We ended up diverting somewhere else, did a few short/soft fields,
then called it a day. If we hadn't called flight following, we would
had never known about the transponder not working, and would have
violated some airspace. Scary stuff. It makes me nervous, as I did
another student's first cross country to class E airport underlying a
class C a few days prior (in a different plane). We made sure the mode
C was on and I even monitored their approach frequency, but I didn't
call them. It really makes you wonder... I just think it's distracting
having to constantly listen for your call sign while you're trying to
explain important stuff to your student.

Anyways, I was wondering, would it be a dumb idea to just call up a
radar facility to just ask if they can see your transponder?

--"Norcal center, this is cessna XXXX 10 miles south of blah blah,
6000 feet VFR, I don't want advisories, but can you tell me if you can
see my transponder"
--"cessna XXXX, sqwak 4545 and ident"
--"cessna XXX yes I see you"
--"ok thanks have a good day"

Does anybody do this? Has a problem like this ever been encountered by
anyone else? Is there any other way to check that your transponder is
working before you enter airspace?