Per what
@375Fox said, they used them enough to mention, back in the day. We know for a fact 303 British was commonly used. Col Patterson favored that in Tsavo to take the lions, among other things. On the USA side, Hemingway favored his "Springfield" which was likely a 30-06 caliber, as he talks about in Green Hills of Africa. On his return safari, he decided to bring something big-bore because he found out the hard way that the 30-06 was far from a buffalo stopping rifle, if even ultimately a killing rifle. It was not the best option but often times, all they had.
Times were different back then. Hunting was basically "best efforts." A lot of animals seemed to be wounded and never recovered. If you read some of the books about hunting Africa, it was basically a free for all in the old days. Tags/licenses were comparatively cheap with multiple animals to a tag. Hunters would shoot a buffalo, go on to track it, come across another, bigger buffalo, while tracking, shoot that one, then break off to track that one.
There are now of course, better ethics and more availability of effective options.
All the true African's knew better. Capstick (despite the controversy surrounding him), Selby, Selous, etc. They all had big-bore rifles for DG from basically the very first availability.