The general history of using FMJ bullets for hunting North American game animals is that it's a really bad idea. The bullets punch through without expanding, and the wounded animal is commonly lost. This naturally ignores a hit to the spine; legs are not commonly the intended target. You can't rely on this "tumbling" thing for effectiveness; there's no such thing as "always".
With solids for large African game, the whole idea is to either break the major bones, immobilizing the animal for a second (or third) shot, or to penetrate the skull of something like the buffalo or elephant in its charge. In-depth penetration is also far more important on a ton of buffalo, compared to a 200-ppound whitetail. And, at 45-caliber and up, they could be likened to "pre-expanded" 30-caliber bullets insofar as blood trail.
Well Art, you're partially right.
I don't know how you know that there is history on hunting with FMJ bullets in the U.S.A. and that it's a bad idea. For as long as I can remember, and I've been around awhile, the official stand has been to use expanding bullets at least when using jacketed bullets, for big game hunting in the U.S.A. In fact, in many places, it's required by law for big game hunting. This ignores the truly long history in Africa where FMJs were use extensively with good results. It also ignores that fact that in many places (probably most) solid lead balls or bullets are allowed for hunting (in muzzle loaders for instance) and haven't been found to be a bad idea; these are nonexpanding bullets. I would agree though, that a tumbling bullet can't be counted on.
It is true that with large dangerous animals, penetration to vital organs is of paramount importance. The same is true when hunting a rabbit, a deer etc. The difference is that it's a lot easier to get a bullet through to vital organs in a smaller and softer animal. Both a solid and an expanding bullet would penetrate a deer and if either hit a vital organ, the animal would die.
The advantage of an expanding bullet is that when it hits an animal, it essentially changes to a larger caliber bullet and creates a bigger wound channel, more bleeding and an earlier death. If you hit a deer with a .224 caliber bullet that expands to .458 caliber you've essentially created about the same diameter wound channel that a .458 caliber weapon would have caused. Of course, the hypothetical .224 caliber projectile wouldn't penetrate nearly as far which won't matter in smaller thin skinned game.
The fact remains, FMJ and other nonexpanding bullets are lethal if you hit a vital organ with them. To postulate that a nonexpanding bullet will just zip right through vital organs like heart and lungs and not do enough damage to kill is a little ridiculous.
Oh, and regards to African game, solids are pretty much the bullet of choice with elephant where brain shots are the norm. Depending on who you talk to, when hunting cape buffalo, a good quality soft is generally the first bullet directed at the hart and lungs. As long as the expanding bullet is built strongly enough to get to the heart/lungs it is better to have a .458 caliber bullet that expands to .600 caliber and is less likely to pass through to injure other animals in the herd. Subsequent shots are with solids since angles including the Texas heart shot or raking shots may be needed and excellent penetration will be of paramount importance.