The click itself is normal, you just don't usually hear it over the racket the starter motor makes. Your symptoms could be caused by anything that makes the motor not run sometimes, including a fault with the motor itself, or even a bad ground to the engine block.
Since you've obviously spent some time on this already, my suggestion would be to connect a voltmeter directly to the stud on the starter (and to the starter housing or engine block). Make the connection permanent enough that you can drive that way, and bring it out so the voltmeter is visible from the driver's seat. I have a 15' length of "lamp cord" with big alligator clips at one end and sockets suitable for my meter leads at the other, which I use for problems like this. Then when the problem happens, you can check the voltmeter without disturbing anything under the hood (or burning yourself on the manifold) to see if juice is getting to the motor. If so, you know the motor is bad. If not, you know the problem is outside.
PS, I once had a similar problem caused by a defective battery (on a non-Triumph). It would test "OK" or "needs charged" using the parts store test, but there was a broken conductor inside the case that would sometimes open up enough to not pass enough current to run the starter. A new battery solved the problem.