Optical audio from Yamaha TSR-700 receiver

I am a new Phonak hearing aid user. My hearing aid connects wireless with a Phonak TV Connector that has optical input. Unfortunately, my Yamaha TSR-700 (from Costco) does not have optical out. ( I tried it's zone 2 RCA out, but it does not give me a stable connection to my hearing aid, the recommendation is to get optical connection).

My current setup has several hdmi input (Roku, cable box, dvd player), and it connects with eARC/ARC hdmi out to my Sony Bravia XR 55X90J eARC/ARC port. I am thinking about getting an HDMI audio extractor like this one ( https://smile.amazon.com/gp/product/B08FR4771P/ref=ppx_yo_dt_b_asin_title_o00_s00?ie=UTF8&psc=1 ).

I am thinking about taking hdmi out from Yamaha eARC/ARC port, input it into hdmi input of audio extractor, and optical out into Phonak TV Connector. I will then connect hdmi out port of audio extractor to eARC/ARC input port of TV for video.

My question is, will Yamaha eARC/ARC port have any audio? Note that I will be using the Yamaha speakers for sound as well, since my wife will be watching TV with me.

Another option would be to move all hdmi input from Yamaha to Sony TV, and use Sony eARC/ARC out port to connect to audio extractor hdmi out port (which is ARC port, I will have to turn the ARC mode on on the audio extractor), and then take optical out of audio extractor to Phonak TV connector, and hdmi audio out of audio extractor (audio extractor has audio only hdmi out) and connect to one of the yamaha input. In this setup I can still connect receiver eARC/ARC hdmi out to input of audio extractor. When ARC mode is set on, the audio extractor should extract sound from TV, but if ARC mode is off, then the audio extractor should extract sound from the Yamaha receiver input. Although in this option, Yamaha does not have any input, but I could play regular FM radio or Musiccast from Yamaha and stream it to my hearing aid.
Thank you in advance for any suggestions about which if any of these two options is better, and weather this audio extractor is a right choice.