Wow! This has gone in every direction imaginable!
1) The statement: ".223 which has better ballistics than all three of them." is pretty wide open. I think if you run the range-tables, different bullet-weights and barrel twists, you will find the .223 is NOT necessarily on top. From the extremes of a 9 twist 20 caliber and a 50 grain VLD, all the way to the 6 x 45 with about 30% more energy than the .223, there is room at the top.
2) The availability of bullets: This can go both ways. Last time I was looking for bullets for an upcoming P-D hunt, I couldn't find the value packs Speer TNT's anywhere, but was able to get bulk packs of 6mm, 70 grain Sierra Blitz-Kings from a couple of different suppliers. Edited to add: The .224 bullet selection is good, but it is also very popular. Often times the popular .224 bullets for P-D hunting will blow-out of stores and distributors early on. The 6mm bullets are not quite as susceptible to being "cleaned out". Bottom line: If the bullets are "out of stock" it doesn't matter how good the "bullet selection" is!
The bullet selection between .224 and .243 is actually quite close, and I have NEVER come close to exploring all of the choices in either one. After all, once you find the bullet that works well for you, why do you need another 99 choices? I have shot the Speer TNT in my .223 for P-D's for about 20 years. I have tried others, but still end up going back to the TNT's.
3) Raw accuracy: Anyone that has loaded for any length of time, and with a wide variety of calibers, is aware of the fact that some calibers are more easy to load for than others. I'm not talking "assembly" of the cartridge when I say "easy to load for", I'm talking easy to find an accurate load for. A load that is accurate today, tomorrow and next month when it's much colder. If you pay attention, the difference that makes certain cartridges "easier" to load for is the proportion of bullet diameter to case capacity. The more "overbore" the case is, the more fickle or difficult it can be to load for.
Take the .308 Winchester case: This case has spawned a whole line of commercial cartridges all the way from the .243 Winchester, to the .358 Winchester. Loading for the .243 Win. is not as difficult as some that I have loaded for, but it is fickle. Step up to the .308 (same case, larger bullet diameter) and the loading gets noticeably easier. Basically, pick a quality bullet, look in the loading manual and pick a powder that will drive that bullet at a velocity that is prudent, and that load will shoot around, or under an inch @ 100 yards. With just a little work, you can tighten that group up. You can find literally dozens of sub MOA loads for the .308 that will shoot that way any day, not so with the .243 Win. The WSSM line that I work with on a daily basis is another example. I probably get a call once a week from a customer struggling with loads for the .243 WSSM, but have yet to get a call for the same reason about the .25 WSSM. The .25 is just that much easier to load for!
All that being said, the 6 x 45, given the same, or perhaps even less attention to detail, will produce more accurate loads than any of the other choices.
4) Between the four choices, three of them feed and function with ease through the AR-15. The .204 Ruger is longer, and with that, can and does have issues with feed and function. When everything is "right" the .204 will run just fine. If there is ANYTHING that is not right, you will be the most unhappy P-D hunter in the field.
I have shot P-D's for several decades with calibers from .17 Remington, on up to the .243 Super Rock Chucker. As far as "bang for your buck" it is tough to beat the .223 Remington with a 50 grain frangible bullet. Sure, you can get more "splash" with the Super Rock Chucker, but at what expense? Sure, the .204 is flatter shooting, but you just don't get the same destruction that you do with either the 6 x 45 or the .223. With the sight-height of an AR, the point-blank range is sooooo good, you really don't need the flatness of the .20 calibers anyway!