Of your choices, for me it'd be between the Rossi or the American (but not in 270). The Browning is a neat rifle....own a model 24 Remington made back in 20s/30s which is basically the same Browning design, that I enjoy shooting on occassions. But, this design 22 rifle is not what I want to shoot paper at range. Prefer my Marlin XT22LR bolt rifle or my Ruger 10/22 for that.
Don't own a Ruger American, but have thought about buying one, just to try it out. Unfortunately, Americans I've found locally use cartridges I either have enough of or just don't care to own. Why I commented about the .270 in above.
As to the Rossi, also don't have one, but did shoot with gent one day at range we are members who had one with him, but it was a 44 mag version, if I remember correctly. Seems it was also suppose to shoot 44 specials too, but he had a feed problem with them, but the 44 mag fed OK. Also, around 10 or so shots with the rifle, he'd have to mess with the cartridge tube under the barrel on the rifle. Recoil would loosen it and it's move forward.