The .308 cartridge commonly used to hunt deer carries substantially more energy than the .223 or 5.56 used in a Mini-14 or AR-15. If not energy, by what metric could the latter be described as "high powered"?
Well the Browning Hi-Power was so named for it's large magazine capacity. But I agree with the substance of what you're saying. 5.56 is not a "high power" rifle if that term is meant to distinguish it from other sorts of rifles. For magazine capacity, you can find 22lr magazines with larger capacities than your typical AR mag, but calling one of those high powered would be a bit silly.
Also .308 is far overkill if you hunt the common small deers, people use it to hunt big game deer, for example moose.
For example this requirement looks like it was explicitly made to forbid 5.56mm cartridges, they are not even mentioning energy:
> The cartridge must fire a bullet with a minimum diameter of .243 inches (same as 6 mm).
http://www.eregulations.com/indiana/hunting/deer/
But for example in Sweden you are allowed to hunt the same size of deers using weapons with half the power of 5.56: 800 Joule.
Only things you need bigger bullets than 5.56 are brown bear, moose and red deer which requires 2000 Joule.
And if you think it is inhumane to kill 50kg deers using 1600 joule bullets, how many joules would you require that people use on a 700kg moose?
Even though Class 2 rifles are legal it's my impression that they have become less and less popular. Most people tend to use something like a 6.5x55, .308 or 30-06 since that gives you more margin of error.
Roe deer is the also largest animal you're allowed to hunt with a Class 2 rifle. The next size deer (Fallow deer) is Class 1, meaning >2000J. You can use shotguns with slugs on boar and fallow deers, but not on Roe deer. This shows that it's not always clear how the lawmaker reasons.
Hunting laws in different countries often reflect tradition and might not always have a sound base in science. In the US you are allowed to hunt with falcons, muzzleloaders and bows. That's criminal in Sweden.
It's quiet and works just fine to kill one. Hunter associations want to cut down on deer that aren't killed quickly so they require larger ammo
You know how it goes "First they came for the socialists..."
* excepting a few that are deliberately manufactured to be subsonic, because they are niche products.
Then there's the urban legend of 5.56 being chosen because injured troops are more of a liability than dead troops
It's similar the the differences between horsepower and torque. To paraphrase a popular meme, Horsepower is how fast you hit the wall, torque is how much of the wall you take with you.