Looking past the Caliber of the Month club - which all the ones mentioned have gone thru - it comes down to where each seems to be positioned in use by owners.
.300BO was invented in the early days of three gun to get around the rules. It was required to use a .30 bullet, and it was because the rulemakers in those days despised the M16 and didn't want it in competition at all. Men were supposed to use "battle rifles" in combat, not poodle shooters. So some imaginative folks wildcatted the .30 by necking out the .223 case and tried it. The rule makers shot it down, but they didn't go away, and eventually the AR was allowed in competition.
It now RULES competition - mostly in 5.56. You can shoot any rifle you want under the rules, and many have tried, but if you want to rank in the top 10% of what amounts to both a precision shooting and timed competition, you shoot the AR15.
Most shoot 5.56 surplus because it's cheap, others attempt to make do with an AK and are hurt less by the ammo as much as the primitive and difficult ergonomics. The ones who get good at it do well. Those with less skills are helped by the AR because it's easier to shoot with less practice - one of the major reasons it's a service rifle and it gets ignored a lot. Look at the modern competitors, tho, and they copy most of the control locations, and for good reason.
As for .300BO, tho, it didn't make the grade. It was a wildcat until AAC introduced it with Remington's oversight, and they were pumping it for a suppressed round precisely because it does well. The larger bullet mass carries more power from a short barrel suppressed than 5.56 does. It's been a major point of discussion the M4 has too short a barrel and doesn't have the M16's long range ability or power from the 5.5" shorter barrel. A .30 cal bullet helps bring back that power in short range applications.
Very few to none appear in three gun circles, tho. Even supersonic, the round costs more than surplus, which is a strike against it for those who practice heavily, and the trajectory is a looper with a lot more practice needed at longer ranges. The 5.56 shoots flatter, less holdover is needed, therefore it delivers more hits on paper with a clock.
Because of those two issues, you won't see many using the other calibers, except x39, which seems to be more a matter of honor than practicality. It's subject to a lot of trajectory drop, too. That makes it an also ran in those circles.
So, at least in competitive circles, .300BO and the others aren't worth the effort. When you are shooting LIVE targets in the field, tho, things are different. And then it goes right back to what target at what range. Again, you pick the cartridge for what you will be doing - not because it's cool or the caliber of the month. Therefore, for LIVE targets, you choose the cartridge that delivers sufficient force as far as you need to bring down a game animal humanely and prevent it's loss or suffering. Any of the cartridges will do it, but each cartridge has it's maximum effective range. Considering the limits of the loaded cartridge length in the AR15 magazine, you can either make the bullet bigger, which shortens the range, or make the bullet faster, which usually lengthens the range. You can't make it both bigger and faster, tho, as there is an offset and the total ballistics in the maximum length of the loaded round prevent it. Plus, the bolt face of the AR won't allow a larger diameter case past a certain limit. At some point you must move to the AR10.
In this conversation, it would then boil down to what the longest range shot might potentially be assuming a 150 pound animal. It's why .300BO and 7.62x39 are used under 250m, 6.8SPC out to 350m, and 6.5 out past 500m. That makes the first two potential close range rounds, the middle an all around performer, and the last a great long distance shooter. That is their optimal use, but in a lot of circumstances, they do ok. You just can't exceed their ability, and have to put up with their disadvantages to get the improved performance they are biased to have.
Most importantly, there is no "best," as they can't do exactly the same thing, so it's always an apples and oranges comparison with no winner overall. Can't be.
. If you are really serious about longer range hunting check out 6.5 Grendel. Dedicated varmint hunters like 6mm/.223 or even .25/.223 (DTI). The great thing about the AR platform is you can take P-dogs with a .204 Ruger swap uppers and hammer hogs with .458 socom in seconds!
Still trying to determine if the felt recoil is any more than my M&P15. 2nd trip to the range and I have 95 gr varmint hand oads doing 2910 FPS with less than a .7" 5 shot group at 100 yds. Once I get some more time, I'll shrink that down some more. 
