when I was younger , my family was not well off and my dad and I shot piles of moose over the years with a parker hale 308 win and 303 British( close to the same). Most times one shot did it. compared to elk , moose drop pretty easy if you place your shot well. Back in those days we just used the regular Winchester PP 180 grain ammo , as my dad was cheap lol.
They are both great calibers, they just aren't as long winded as some other calibers. But when you use them at a reasonable distance, they work well.