The .303 British (not to be confused with the old .303 Savage) is perfectly adequate for any game animal in North America. I would avoid British military ammo, which is corrosive, erosive, cupronickel jacketed and Berdan primed. There has been some good ammo on the market lately and of course U.S. commercial ammo, although it is underpowered because of the old Lee rifles. Reloads in a good Rifle No. 4 can produce surprising results.
Many thousands of Rifle No. 1 Mk III's and Rifle No. 4's have been sporterized, either commercially or by individual gunsmiths.
Today, it seems almost a shame to destroy part of history, but those rifles are so common that I suspect the "chop and channel" jobs will continue for many years.
Jim