I have two Uberti lever action rifles in .38 Special (an 1866 repro) and .357 Magnum (1873). I've been shooting cowboy action for seven years and have put quite a few rounds through these rifles.
I don't use round nose bullets in my rifle ammo. I have shot a few semiwadcutters, which as luck would have it fed well, a fair amount of RNFP (round nose flat point) bullets, mostly 158s, and several thousand rounds loaded with TCFP (truncated cone flat point) bullets. These are my first choice in lever action rifles and are available from some ammunition manufacturers, including Georgia Arms.
Your Winchester '73 is a really nice rifle. Enjoy!