This is why I've always believed in having more gun than what you need. If you take a far more powerful weapon, it gains versatility over a weapon that is marginal for its role, a sloppy over exaggerated example would be how a .44 Magnum can be reduced down and still be capable of self defense or small vermin, while a .380 is always going to be run red hot to maximize performance. I reduced reloads for 7mm Rem Mag to reduce recoil and gain accuracy in the rifle, and it still had more than enough killing power left over for deer at distance. Someone reloading .243 for longer distance deer hunting might find themselves less likely to cool down the load, because they might be more apt to want every bit of power they can get. We could argue about .223 being adequate for deer in the best conditions, with a short barrel and a cool load making things even worse.
It is a very broad question, but good for raising thought, I suppose. In the case of self defense, hot as one can run is the best, i believe, because the handgun is already under powered,and inherent accuracy not an issue at close range. In hunting with a handgun, it is a very tricky issue, because you want power and the greatest accuracy possible, meaning trying out various factory and handloads is well worth your time to attempt to maximize both. For the rifle hunter, I think I've already explained myself.
The other issue of power is trajectory. My Tommy gun is actually accurate at 100 yards, but the lack of velocity means Kentucky windage comes into play at a short range for a long gun. A lower power round may be more accurate, but a slower bullet will have more adjustment at longer range for drop and travel time. Even if the hotter round is a little less accurate, perhaps it shoots a bit flatter, helping out with shots if you lack a rangefinder.