I find it interesting that so many hunters in the UK seem unfamiliar with the .303, I would havethought a great many of them would have grown up with them as a first rifle like over here in NZ. It used to be the predominant rifle. Nowadays you dont see so much of them out hunting narly as much, for one thing they are getting long in the tooth, but also because new rifles have got so much cheaper to buy today than ever before.
I have shot big stags with a .303 and they have gone down in the same way they have with the .30/06. I would equate the .303 with a .308 Winchester untill you get up to the heavy bullets. A 215 grain bullet in a .303 is a different story.
Plainly you could hunt the world with a .303 loaded that way and get by just fine - and people did for decades. The .303 Lee Enfield .303 would have to one of the most successful hunting rifles and cartridges ever made, used worldwide on all kinds of game on every continent for most of the 20th century.
To answer the OP, it was themost widely used cartridge for deerstalking for many years indeed. It is now effectively obsolete, although there are still a ton of old rifles floating around. .303 ammo is every easy to get over here, every gun shpp will carry some, but even in the last twenty years I have seen that its use has dropped right away, again, due to cheaper new rifles being available. You dont need to start with Dad's old three-oh anymore when you can buy a new Zastava, Howa or Savage so reasonably.
Brithunter, just out of interest I was perusing old copy of the Kenya Gazette and reading all the listsof people reregistering their rifles in 1908 - it is plain that the .303 was the most popular modern medium caliber, and it was interesting to note that half of the .303's were chambered in Mauser rifles. (They would have been Obenrdorff made rifles at that time, but years ago people used to get army surplus Mausers rechambered and set up in .303, simply because .303 ammo was so cheap. Used to sell it by weight, scooped out of barrels.)