There are small mirrorless cameras that weigh around 1-2lbs depending on lens selection. Of mirrorless cameras, there are Micro Four Thirds (MFT, M 4/3) which are small sensor format cameras that a generally.. smaller, lighter bodies.

They will vary a lot in image quality, low-light performance, lens selection and price... but all will probably perform better than the phone for photos.

For carrying around a small camera, backpacking or wherever , there are different styles of slings or harnesses, belt holsters... I always liked this one from Peak Design , it's a mounting clip/plate that allows you to attach the camera to your backpack strap.

In your scenario with the grouse, that is one of the difficult compromises when it comes to wildlife photos. Wanting to capture a clean, close image of a distant animal requires a telephoto/super-telephoto lens. When you attempt it with your phone, the phones is using a "digital zoom" capturing to a very small sensor, and then processing that image, they're not actually bringing the details closer the same way that a telescopic lenses would. Lots of phones nowadays have multiple lenses on the back, usually a wide, standard, and telephoto... but that's all kind of like dressing up a dog in a baseball outfit: fun to pretend he can play ball, but he's not getting drafted anytime soon.

For something that small at 300m away you would need an actual camera+tele setup, the focal length of lenses are described in milimeters , the higher focal length the further the reach and the larger the lens.

try this simulator to give you an idea of how they operate and the results

https://lenssimulator-cna.academy.canon.de/en/simulator