As others have pointed out, most safeties lock the bolt, so it's not going to be pulled open. My left arm is also shielding the action in this position. When going through brush, I'm also turning my body to avoid dragging on limbs and such. If it's really thick, the rifle is in my hand, with even more control. I've never been in the habit of using my rifle to part the brush to get through either.I, too, like the Rhodesian carry but with one caveat....it makes it SO easy for brush to snag the bolt handle knob and open the action--have had that happen several times since a Mannlicher Schoenauer is so slick and easy to open. The rifle, scope side down means that the handle is turned forwards and acts like a brush hook.
When carrying a bolt with the muzzle up, I have had the bolt handle catch on my clothing, and the muzzle can get hung up in low hanging branches as well. That's a big reason why I started looking for a better way to sling my rifle years ago.
Tried a sling called the Safari Sling, which carries the rifle horizontally in front. Very easy and quick to use, good muzzle control, sucked in the woods. It made it a bit of a pain to navigate even moderately thick timber as the rifle really wants to be almost sideways while carrying. Works great in open country, just not so well in the thick stuff.