Early in WWII, our War Dept. had an uncharacteristic moment of insight. They realised that:
1) there were a lot of guys who were not front line combat troops.
2) these guys did need to be armed. Traditionally this meant the 1911A1 pistol, rather than a combat rifle.
3) They were not going to have the time, nor the ability to make most of these guys decent pistol shots.
4) a light carbine, with neither the power nor the range of the main combat rifle would fill the bill, giving supply and support troops something that they could actually use effectively at short ranges, on those rare occassions when they needed to.
The result was the M1 .30 caliber carbine. It was never intended to be a combat rifle, however, when the combat troops got their hands on them, and found a rifle that only weighed about 2/3 what the M1 Garand did, held 15 shots, as opposed to the 5 of the Springfield, or 8 of the Garand, and performed tollerably well at close ranges where most of their combat was happening anyway, they used everyone that they could get their hands on.
Lighter and more range than the Tommygun, too, if not the knockdown power...
later 30 rnd mags just made the carbine even better.
The .30 Carbine round is approximately equal to the .357 Magnum pistol round, except that it cannot be used with heavier bullets than the 110gr standard. From a sporting perspective, the .357 in a carbine is more effective, but the carbine round, with soft point bullets will suffice for deer at close range. Compared to other ".30 caliber" rounds the .30 Carbine is the least powerful, and not as good a choice for hunting, for a variety of other reasons as well.
The only thing the carbine round has in common with the .30-30 or the .30-06 is the diameter of the bullets. Bullet length, weight, and velocity is much greater with the larger .30 caliber rounds.
The .380ACP (9mm Kurz) the 9mm Luger, and the 9x21, 9x23, 9mm Largo, 9mm Glisenti, (and some others not mentioned) all use the same 9mm diameter bullets, but the different size cases deliver them at different speeds. The .38 Special and .357 Magnum use .35 cal bullets (.358 vs the .355" for 9mms), but again, case size and pressures deliver vastly differing results.
Other than a single model of the Ruger Blackhawk, and single shot barrels for Contenders (and a single long defunct attempt at a semi auto) and the slightly successful Automag III auto pistol (also now long gone), nobody has ever put the .30 carbine round into any gun design that wasn't a variation of the M1 Carbine design. Its a nice piece of history, great fun to shoot, and does have some limited practical uses, but its not a general use rifle, and was never meant to be.
All else being equal (and it almost never is) bigger bullets tend to work better.