I stream practically to the CI only because of my willingness to rehabilitate more effectively. If I cannot understand, I switch to phone speaker or hearing aid side only (Phonak Paradise 90-RT).
Theoretically, with a Cochlear Phone Clip (an intermediary Bluetooth Classic device), I can stream binaurally, but I found it cumbersome. I haven’t yet found a way to get binaural streaming with Bluetooth ASHA to CI and Bluetooth Classic to HA.
The Phone Wireless Clip battery lasts 5 hours now (8h after receiving), after heavy use (IIRC, Ni-MH), so recharging it is also inconvenient. The Nucleus 8 sound processor is definitely more convenient with direct streaming, which translates to more hours of rehabilitation.

what eventually got you to isolate the CI ear and how long did it take for it to start working for you? I’m sure I remember correctly that you do well with it now.
I bought a Cochlear Wireless Phone Clip (described above) and it stubbornly and consistently streams only to my Nucleus 6, for example, when I return from work. Streamed a lot whenever possible, because it’s basically the only way to isolate my better ear.
I requested a custom silicone earmold, which functions as an earplug specifically for my better ear, but it only offers approximately maximum 40 dB of attenuation, so it is not perfect for isolation.

how long did it take for it to start working for you?
Hard to say, because although I didn’t regain usable hearing during 2013-2016, it provided some foundation for further training after the subsequent 6.5-year break.
I gave it a try because @Raudrive, in private messages, discerned between the lines that I was dissatisfied with my results, and he recommended that I isolate my better ear… IIRC, this was in March 2023.
After consistent heavy streaming - I need a few months to achieve reception of sentences, but not perception yet. See screen below - I mean “Identification” and “Distiction”.
Now I am gradually transitioning to “comprehension” and “recognition”.
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See my post; it describes WRS in January 2024, ten months after @raudrive’s advice. During that time, I had a three-month gap due to studying for an important exam—keeping CI working was exhaustive…
It’s my opinion also. Almost year ago, in Apr 23’, @Raudrive have had given advice to hear only by CI on my left without any involvement of my better ear. I used my Phone Clip for Bluetooth streaming extensively to listen to podcasts, even if I didn’t understand them at all. I had a break from rehab about 2,5 months due to exam in October, after which I returned to CI training. At the end of January, in WRS test in quiet (CI only in 65 dbA) I have 80%I had not perform that test on …