I have Oticon OPN S 1 aids that are paired with my phone. I also have the current version of the Oticon On app installed (version 2.0.3).
Control Center is part of iOS, at least the later versions. If you start Setting in iOS (12.4.6 on my older iPhone 6 Plus), then select Control Center (under General on my phone), you can turn on the Access Within Apps setting.
You access Control Center by swiping up from the bottom of the screen. With the Access Within Apps setting on, I can access the Control Center from within the Oticon ON app.
If I swipe up on my phone screen, in the upper right part of Control Center there is a Music section with a small blue icon (small triangle at the bottom with 3 circles) that allows selection of the sound output device. On my phone I am allowed iPhone, Stuart (hearing aids) or Living Room (a Sonos sound bar) where my hearing aids are selected by default (check on the right side).
In my version of the ON app in the remote control screen, there is a small icon centered at the bottom of the level slider for each program. It looks like a speaker with two rings in front. If I click that, the icon changes to a speaker with an x in front and the microphones in my hearing aids are muted. Click the icon again and the mics are turned back on.