Help with a dumb smart thermostat

Ok, I don't think the problem is actually the thermostat in this case, but I've gotta be all get-off-my-lawn about something!

My friend has a fairly standard furnace/AC unit on which we installed a Nest thermostat a while back.  It had the regular fan, heat, and cooling wires connected as well as the C and Rh wires.  It was working fine for a year or two until the end of last week when it started giving the "delayed for xx:xx" message.  From everything I can find, that means that the internal battery is low and it needs to charge itself to avoid over-cycling the compressor.  The most common cause for that was not having a C wire hooked up, according to my perusal of the internet.  At the time, I didn't remember what I had done when I installed it and I didn't have time to go over and look at it, so I told her to order up a 24v AC-AC transformer.

I went over today and installed it and found that there was already a C and an Rh pair connected, coming from the unit, right off the control board.  Okay, weird, but maybe it's just not enough voltage or...  hell, I don't know anything about how this is supposed to work but I've got this transformer, so what the heck.  I disconnected the power coming from the furnace and connected the transformer to the Nest.  When I plugged the thermostat back in, I ran through the system test and it was able to turn the blower on and off, but then did not turn on the A/C when I tried the cooling test.  I popped the breakers off and back on, and still nothing.

Connecting the C and Rh wires back to the furnace's wires and removing the add-on transformer, everything works fine.  Swap everything back and the system will turn the blower on and off for the Fan test, but then nothing else.  Come to think of it, I didn't think to try a different test first, but that would have been an interesting data point.

I was under the impression - and I freely admit it may be an incorrect one! - that the line coming from the furnace to the thermostat is power delivery only?  Is it somehow backfeeding power from the Nest to run the furnace controls and draining the battery in the process?  Is there an HVAC electrician in the house that can illuminate me?

edit:  okay, so as I think about it, the three wires going to the control board are isolated here and I don't have any thing from the furnace to complete the circuit, so of course that doesn't work.  Am I supposed to double up the transformer and the Rh?  If I connect one to Rh and one to Rc, it seems like the Nest will jump them internally anyway....

I'm guessing a broken wire from the thermostat to the furnace.     

Just pull a new one.

The C wire is actually a return line not power.  On a traditionally wired, dumb system there's 24vac power going out to the thermostat on one or more red wires and the thermostat sends power back to the green wire to turn on the fan, white wire for heat and yellow for the AC.  A smart thermostat needs a return or ground wire to charge the battery and keep the display on.  That's what the C wire does.  There's no official color for the C wire but it's typically Blue or Black.

With the thermostat removed you should measure 24vac between the red wire(s) and the C wire.  If you jumper the red to green the fan should run, red to white the heat should run and red to yellow the AC should run.  If all of that happens then you've got an issue with the thermostat itself.

All of that assumes that the installer followed the color convention.

Had this same issue with mine this past year.  I first replaced the thermostat and thought everything was good until we got the power bill and discovered we had been running on emergency heat the whole time but the unit just displayed heat.  Couldn't figure it out so ending up calling for help. Ending up replacing the thermostat again, outdoor control board and contactor.  Not sure which one was the culprit or if one knocked out the other.