The real issue is choosing a cartridge before the game. It's doing it backwards.
Choose the game, then what ballistic envelope you need to hunt it. What range, what target. It's obvious when we discuss deer out to 200m vs prairie dogs out to 500m - there may be an overlap in performance but few would say the same gun would be optimal for both. Even the optics won't cross over well.
Deer can be seen and hit out to 400m, if you choose a cartridge that only goes half as far are you knowledgeable before hand that it's lacking? Goes to understanding what range and target. Deer DO need 1,000 foot pounds of force to ETHICALLY drop them close to where they are shot. A lesser cartridge won't do that and mortally wounded deer could get away. It happens all the time and is frequently posted in hunting subforums. People sometimes spend DAYS tracking them out only to discover them too late for a food source.
It's why game departments do make arbitrary decisions on what cartridges are legal.
MO had a minimum caliber requirement at one time but as the average hunter matures they understand that the smaller bullets tend to have a shorter effective range and hunt accordingly. I did the same - .308 at first, buddies were using up to 8mm Remington Magnum. As time wore on I kept seeing that the effective range on Ozark broken woodlands was actually only as far as I could see IN the woods where deer would move in daylight hours late fall. That was 200m max. Even then, plotting all the known kills in the area I hunted for the last 40 years, I found most of the shots were under 100m, usually less. I didn't even need 6.8, which I had moved to.
Taking advantage of the MO seasons I found I could hunt twice as much with an AR PISTOL in 5.56, which has 1,000 foot pounds out to 80m hunting exclusively on wooded ridgetops where most of the daylight traffic occurs. I built one and used it last seasons - plural - knowing it was "enough gun" for any deer I could see between the trunks of the trees.
You could use .300BO just as effectively, but you can't shoot any cheap surplus in it, or 6.8, or 6.5, regardless. They are commercial loads, not military surplus, and there is no cheap ammo. Period. 5.56, yes.
Choose whatever tickles your fancy, but if it doesn't fit the requirements for your hunting game for range and target it's not going to do the job. Don't choose the cartridge first, examine the hunting requirement and then choose a cartridge that FITS. I was never a great fan of 5.56 back in the day, and the AR wasn't my first choice of rifle. Using one for 22 years in the service and then building one I began to understand it. Using the cartridge in the service for 22 years and then building an AR pistol I fit it to the job that I needed done.
Is .300BO a good choice? Maybe - but if you work thru the range/target question it's a more sure choice than simply going with the Bullet of the Month and then hoping it works. Will it deliver 1,000 foot pounds out as far as 85% of your potential shots hunting? is the real question. We are all presented with game shots hunting but not all of them are actually high percentage situations - go for a choice that gets most of them and you will know which ones would have taken a crew served weapon and forward observer.
Range and target first, then the short list of what conforms to that ballistic envelope. That's the correct choice.