Personally, if the session is important enough to bring a big camera then I'll just deal with the drawbacks of schlepping around a big camera. If all day comfort or equipment safety outweighs importance then I tend to just commit to snapshots and maybe one real "photograph" via cell phone if it works out.

Then again, I'm shooting a very light Canon T5, and when paired with a 50 1.8 it's remarkably light.

I try to stay streamlined as much as possible, and I use an over the shoulder strap just like that, but I leave my tripod adapter attached at all times, and the retaining screw has a little metal loop that lets me hook onto the camera. That way I'm always ready to strap up or clip into a tripod without fumbling for gear.

For wildlife walks, I try to hand hold and use the strap to carry the tripod (slightly extended to remain balanced), but I'm also using a kit 75-300 and not heavier, upgraded glass. A hand strap like this could be helpful to reduce strain with heavier lenses or bodies.

Wildlife is really challenging me lately (and I'm beginning to feel where an upgraded body with more control like you found would be very helpful). I'll usually dial in a manual setting exposing for the sky for birds in flight so I don't have to bother grabbing an exposure; then I'll switch to shutter priority for the rest with a high shutter speed to mitigate camera shake. An A and B memory setting would rock.