You're asking about chamberings, not action types or rifles....
Some of the better,
reasonably-affordable (i.e. non-NE calibers etc.), chamberings for the
largest DG that you mention, in my understanding, are those that perform well (meeting the standard minimum velocity and bullet weight requirements, such as "400 gr @ 2400 fps", etc.) for your tastes and preferences, AND are large enough to be legal in the particular country you're going to (can be "over .400" caliber or a certain energy requirement, or weight plus velocity requirement), AND (perhaps most importantly if you cannot afford a double rifle),
are a relatively LOW-pressure round, so that they will not bind up the action, but will rather extract reliably in a turnbolt even in the hot african sun to save your ass - they include some of the tried and true rounds like:
-.404 Jeffrey
-.416 Rigby
-.505 Gibbs
-.458 Win Mag (and .458 Lott)
All of those have a LOT going for them, especially the first two. There's a few others that meet these requirements, that I'm forgetting, I think. The bolded concern is why the really "hot" stuff is not more popular (.375 RUM, etc.), in my understanding. Personally, if I get a true African rifle, I'm leaning toward .416 Rigby or .458 Lott. I figure my 9.3x62 will handle up to Eland.
That's DG. As for Non-DG ("plains game"), just your standard North American chambering of choice will cover everything (stuff like .270 Win, .30-'06, whatever) you will encounter, until/unless you go up against something really big like Eland (or maybe zebra), where you might want a .338 Win Mag, .35 Whelen, 9.3x62, .375 HH mag, etc. Those last ("medium") chamberings are also popular, in my understanding, for smaller DG like Lion and Leopard. I believe SOME countries allow the .375 HH mag as an *exception* to their general "must be 400 @ 2400" or whatever rule, for large DG also, because it's considered such an effective, tried and true chambering, even though it's not .400 or larger! So
if you go to the right place, and
if you want to take "just one" rifle (for some reason), then the chambering should be .375 H&H magnum.
You can hunt just about anything legally (AND morally - not endangered species) somewhere in Africa, if you have enough $$.
Going on a multi-species non-DG plains game hunt is actually quite affordable right now,
relative to the past, and
relative to alternative out-of-country trophy hunting trips like Canada, etc. There's a lot of competition for your safari dollars, and that's good for hunters. At least it's relatively cheap BEFORE taxidermy fees, for said MULTI-trophy hunt.
Oh yeah, if you want a true african DG rifle, but yet in a very affordable package, then the chambering for you is .458 Win Mag or .458 Lott (or in some circumstances .375 HH mag), and the rifle for you is the CZ 550 American Safari Magnum - you can get a true DG caliber in a sub-$1,500 rifle, IINM, because the action is slightly smaller than "true" DG actions. BUT, beward cracking stock concerns on CZ 550s.