They did. Its called .243 Winchester. A high performance cartridge--100grain bullet with muzzle velocity of 2900fps. It is a necked down .308/7.62 case with a 6mm bullet. Short maximum length of 2.7-in.--ideal for automatic weapons. Light recoil. Women and children have been knocking off white tail deer with this for years.
I would like to see what American arms designers could do with such things as gas-pistons(AK, M14,), or delayed-recoil(Browning M2, Thompson) and a cartridge such as the .243.
BTW, I also don't think we need all the geegaws we seem to hang or attach to the rifle. What we need is something which follows the concept of RAM-D(Reliability, Availability, Maintainability, Dependability) first and only then play with the toys. Weapons following this concept well are much liked. Witness the affection of older vets for the M-1/M-14 rifles. It's near universal. The M16 has been "controversial" for 40 years, and is still viewed with suspicion by a great many who use it.
This alone should tell us something about what to replace it with.
And the Navy tried several M14s in .243 for SEAL team used, and came back unimpressed, largely due to problems with the rifling twist of the barrels, as those suitable for long-range work are less suitable for up-close work with more lethal projectiles. Neither were barrel luives of 10,000 rounds considered sufficient in weapons meant for fully-automatic fire.
The British .280/30, originally meant for their experimental EM-2 rifle of the 1950s, might have been another step in the right direction. But I suspect we'll be stuck with the M16/5,56mm cartridge combination for so long as conventional mettallic-cased cartridge ammunition remains state-of-the-art.
But when caseless or plastic-cased ammo comes along....
Armalite is producing a AR-10T model in .243 ... you might be interested.