>caliber war African Boogaloo
Despite having little hunting experience, I did do my fair share of reading up on safari hunting and have talked to a few people that I know to be knowledgeable about firearms/hunting but also know not to embellish stories; some have given me interesting insights into the safari scene.
One particular gentleman conducted a cull as it's one of the lesser expensive safari hunt options. Since he wasn't out for dangerous game, he simply flew over there and rented out one of their rifles, a Ruger bolt action in 300 Win Mag, that he used to take 11 animals with 12 shots; the last being to finish the creature off after it had been anchored. Although he had a blast, he warned me that doing a cull will not get you much love from the guides as they aren't really making much money off it, but dangerous game is beyond the price range of most mere mortals.
Even then, most safari hunters and guides/owners of safari hunting grounds don't commonly reload when they are on a hunt. They tend to go with a quality box of factory ammunition; keep in mind that people who can afford to hunt big game might simply not care that they can reload their .500 Jeff for 1/10th the cost because well, they don't give a shit if something is 50 bucks or 250 bucks, they don't have pocket change that small. The added benefit is that factory ammo is "safer" in how factories making premium ammo tend to not fuck up as often; your reloader might be very careful but one little mistake and people can get trampled. No good for business.
So you end up with all these hunters over there following one fairly a few simple rules of thumb:
Bonded and/or partition if not for dangerous game.
Solids for dangerous game.
Minimum of 375 H&H for dangerous game.
After that, one of the big things is going with a "common" reliable caliber that has already been used fairly often with good results. This doesn't invalidate any other caliber, but you have to remember that those who organize the hunts aren't going to stock every fucking caliber on the face of the earth. If you use your own rifle, they will most likely want to know it's not going to have a stoppage and get someone killed, and being able to buy ammo from them could be important for multiple reasons (they want you to use something they know will work, even if you know it works, they don't want to take the chance that you may have brought in a dubious handload, etc). This can be compounded by some import/export laws that could very well put a wrench in your plans of taking your AI 8.58×70mm Military McTactical Police edition through whatever fuddington bix nood customs.
This brings me to another point. Most .338 Lapua factory loads are aimed at long range shooting with possible armor penetrating purpose. Are those bullets going to work well for really big game? You might think so, as they have a high sectional density and a bunch of power, but penetrating YARDS of flesh is nothing like penetrating armor meant to resist bullets. Long range bullets tend to be pretty ass-heavy and pointed, this is generally not as good at going through multiple feet of meat, as it will go sideways very quickly. It may penetrate deep when ass-end forward, but at that point it might not be going in the line you were aiming for, having veered off. Solids aimed at going through walking flesh tanks are shaped like flying bricks for a reason; they go straight, fuck you, suck my dick, I'm a brick. Cylinders of copper or brass just go straight like they're on rails, punching through what you aim at, even if all manner of biological tank-ness are in the way. For not dangerous game, good hunting soft-points are fine; but this brings me to a further point.
How many factory loads designed for hunting are there for the .338 Lap? How many hunting-configured rifles are there in .338 Lap? An "inferior" 338 Win Mag rifle will be lighter, more portable, easier to shoot, have tons of quality hunting factory loads, and the hunting grounds will have more than one box of ammo on site. Even if handloaded or specialty 338 Lap could be found with dangerous-game designed solids and soft-points that are made to handle the extra velocity of the Lapua at closer ranges, will the guides trust your handloading or trust a box or ammo they haven't seen before to ensure their customer or employee won't get used as fertilizer? Will you be okay with dragging around a rifle with the weight and bulk of a .416 Rigby that shoots .338 caliber bullets, and doesn't meet the perhaps-arbitrary-but-still-used-375-diameter-minimum-rule?
6.5 and 7mm loads used on elephants were also used mostly in the era of round-nose, heavy non-boattail bullets with pretty solid jackets.
I'm sure you're going to find articles of 338 Laps used in safari hunts, and 338 Lap hunting rifles, and 338 Lap hunting loads with softpoints good enough and solids in the right design to post in my face and tell me I'm dumb, but the main point of this whole somewhat anecdotal and heresay-filled rant is to try to keep it simple and relatively cheaper when going for a hunting expedition that's already fairly complicated and expensive when there are plenty of proven, common, well-known guns that are made for the job, and most of all that the guides will have familiarity with and will trust.