Rapier or broadsword. Both work, and neither work if the bullet is in the wrong place. As the client, you have one overriding responsibility - put the first bullet in exactly the right place. Do that and everyone goes back to camp to celebrate your bull. Blow that first shot and very bad things can happen and likely not you.
For a first buffalo, I would strongly recommend the rapier rather than the broadsword. If you have done a bit of deer hunting, then I suspect you are very comfortable with a scoped bolt action rifle. The easiest transition is to a .375 of similar configuration. Load it with a quality 300 gr bullet like a Swift A-Frame and it will kill any buffalo that ever lived just as dead as a .416 or any other forty something.
Scope it. On a thirty day safari to East Africa in the fifties, you could play around with open sights, heck even wound and lose a buffalo or two and no one would care. On today's typical 7-10 day buffalo hunts your one and only shot at your trophy of a lifetime may be a old bull standing seventy yards away with three of his pals in very dark brush. The shot window to your bull is a foot in diameter. With a scoped .375 off the sticks that is an absolutely makeable shot because you can clearly differentiate your bull from the others and see the shot window. With open sights, or worse yet, a double with open sights you will see a black blob and get to test your ethics as a hunter of whether or not to walk away.
Secondly, the .375 is the most versatile rifle you can take to Africa. By that I don't mean it is a good compromise. Rather, no one has yet created a more versatile chambering for everything that walks on the continent. On one typical buffalo hunt in Moz, I took a great bull, and a great selection of plains game ranging from an 8lb Suni through sable and Lichtenstein Hartebeest to a two-thousand pound Livingstone Eland. All were taken with the same rifle, load, and scope.
I own a .404, 500/416, 450, and .470. I shoot them all very well. All four of my buffalo were killed with a .375. All were killed with one shot, though two were given insurance shots after they were down, and none went more than 25 yards are so. When it is time to pack up for another trip to Africa, it will again be a .375 that accompanies me. Nothing is more versatile.
