I had a P0014, that triggered my TC , what caused it , loose timing chain! Yep, pretty light! Puts the fear of God in my wife, she came home bawling her eyes out, Thought she had killed the BlackSwan, 5 hours later , all good!
she didn’t pitch in and help with the wrenching or clean up, but she did fix me a nice cherry cheesecake! My favourite!

 

Well I learned a bit. I've been retired for several years and my experience with these is limited. I'm an active member of iatn and asked the guys over there.
My experience with T/C and engine problems was if you had an engine problem that was noticeable and or CEL is on you fix that first and it may return the T/C to operation. Over at iatn they say now there are some manufacturers like Toyota that a loose gas cap will illuminate every light on the cluster.
As for open loop causing the light that is not exactly correct. The consensus seems to indicate that an engine problem that is severe enough to turn on the T/C light is likely going to put the PCM into open loop. It's more like open loop is caused by the same thing that turns on the T/C light. They do not effect each other directly but rather indirectly.

 

Thing is most people would never notice a problem in open loop, without some sort of notification. 90% of drivers only know how to open the fuel filler cap, not even that in New Jersey. GoOgle "what does the red light on my dashboard mean" amazing how many pages of results.

For those that don't know; it is illegal to pump your own gas in NJ..

 

New Jersey is funny for those who don't live there. LOL