After getting frustrated, I took a day off, which for me is sometimes a good move, especially when it is going to be weeks before the ice clears from my driveway.

DEcosse was correct about the lamp socket. The LEDs I bought have bases which are more square than the OEM bulbs, I thought it was just a tight fit, but it was the contact 'mashing' to the bottom of the socket. The fact that this was even possible speaks to the quality of the design.

I took the sockets apart, straightened the contact, corrected the polarity (3 out of 4 were backwards), and carefully put the new bulbs back it and they flashed, much to my relief. After tapping the signal wires under the right side cover (soldered T-Splices) and connecting in 12 ohm resistors (between the green/red and ground and green/white and ground) to correct the blink rate, I now have my indicators back.
Since I was on a roll, I also put LED bulbs in the tail/brake light, license plate light, headlight, and driving lamps. The only difficulty there was that the OEM driving lamp harness is also wired backwards, with the white wire connected to ground and the black to the relay.
As to facts, I prefer as a general principle to be able to check them myself.
- It is a fact that the relay under the side panel is the starter relay and not the flasher relay on an R3T even though it is labeled that in the service manual. Don't take my word for it, check the wire colors leading to it or just pull the relay and if your bike starts, I am wrong.
- It is a fact that a 'Diode Pack (indicator)' is shown on the circuit diagram outside the instrument panel in exactly the same place as a flasher unit would be. The service tech at Triumph USA confirmed that the diode pack exists, and indicated that it is wrapped up in the wire harness somewhere. You may be right and the flasher unit may be inside the panel. I still suspect that the instrument panel has a timer loop which shuts off power to an external flasher unit, making it cheaper for the dealer to fix a blown flasher. Either way, my lights work now, so it is not worth cutting up my harness just to do the measurements and find out.
The real lesson I've learned from this exercise is that the idea behind 'Measure twice, cut once' applies to Triumph electrics as well as carpentry.
My advice to any who have read to this before installing LED turn signals: Get a meter and verify that the ground connection on the side of the bulb socket is connected back to battery ground. Save yourself a bunch of time and frustration.
Thanks to all who replied.