I have shot elk with 270's, 7mm's, 300's, 340's and 338's. I have seen it where the 7mm and smaller could require a second shot. A couple needed tracked down but that was due to a bad choice of bullet for the distance. That is all I will say about that.
I feel the 300 win is better than the 7mm and smaller chamberings for elk. That is why we shoot a 300 win. now. We are elk hunters. That said the 7mm down to 270 and even 25-06 will do fine on elk with good placement. But I feel the 300 will do it better and for sure farther as a normal rule. Why?? Bigger (heavier) bullets.
Any of these rifles (7mm, 300, 338) I would want a recoil deduction of some sort. A good side discharge muzzle brake is what I prefer and foam plugs in my ears for the field. It only takes a second to put them in, although we seldom use the ear plugs out in the open.
I use the Bergers and have for years. We have never tracked an animal with them. They have covered our butts on occasion with shots placed back too far. That is where the fragmenting bullets shine. You will want to use the larger offerings for the choosen caliber. We shoot 210's 215's or 230's from the 300's. and 180's from the 7mm's. If you are up close they may not exit due to the velocity and massive expansion. But rest assured if you are close to the vitals area all vitals will be destroyed. As far as getting fragments in your food. NO, never, not even once. We cut out that area and toss it away. Just like we do with blood shot meat from the impact of any bullet. I do not feel we loose any more meat with a Berger than we would with any bullet. I fell velocity is much harder on meat than fragments. Plus I use what I feel kills fastest. No meat loss from a bullet will ever be as large as loosing the entire animal because he ran off with your bullet.
I think you might want to read this thread I wrote. In it you will see where I tested the 215 Bergers from my 300 win last season. There are pictures of entrance holes, exit holes, and internal damage. Also there are distances, shot placements, and impact velocities. We took 19 big game animals last fall all with this rifle and bullet. Some small as antelope, some as close as 200 yards some as far as 1285 yards and several elk too.
Please notice I am 54 years young, as is my wife. Also note we took young hunters as young as 13 for their first bull. We also took out 3 hunters that were from 62 to 73 years young. Not one complained of recoil or noise. It is a long thread but I think it will be worth your while to read. It contains answers to some of your question that you will be able to see and draw a conclusion for yourself..