Got it.I was ready to order one of these last night but decided to wait until I tried getting audio through the stereo with the optical connection from the TV and I haven't succeeded yet.
I'm too embarrassed to admit why it wasn't working.
So now I get TV sound through the audio system when it is on, but through the TV speakers when the audio system is off. With the optical-audio splitter Mitchell has recommended I should be able to get the TV sound through the audio system speakers and directly to my hearing aids simultaneously.