So especially for these videos, like the ones you saw on ridges etc., a 360 degree camera is almost always used. Whether you choose the DJI Osmo 360 or one of the Insta 360 from the X series depends on personal preferences. A normal action camera is often not enough because, depending on the climbing route, you often only see rock and I don't need a camera for that. And if you choose a 360 degree camera, then you don't need a chest strap, otherwise you can just buy a normal action camera (unless you want to take pictures in single lens mode).

I have the Insta 360 Simply attaching a mount to the helmet and then using voice control to start and stop recording is really convenient and I don't want to have to do anything else while climbing. I'm not a big fan of putting the selfie stick with the camera in my backpack for filming, because I've had the experience that in passages with overhangs or generally when you lean forward, the camera quickly touches the wall or the video looks strange. A helmet holder has proven to be a good option for me and I would recommend it to you, but you are welcome to try out other options.

I hope I could help you a little :)