This gets us into the area of tradeoffs again. Since WW1, it's been accepted that the full-power rifle round was more than the individual soldier needed, or could use. Aimed rifle fire beyond 300m was rare, but still the price was paid in terms of a more powerful, more expensive round, and a longer, heavier weapon. The Germans pretty much proved that in WW2 with the MP44, firing the 8mm kurz round. It was the world's first assault rifle (sturmgewehr was Hitler's own term, adopted after he finally approved the concept), and it was so good at killing Russians that the USSR started a crash course to develop their own equivalent.
In both cases, they stayed with a lighter bullet, but still the same diameter as their full-size round, 7.62mm for the soviets, 7.92mm for the Germans. It was a good, safe move, and served them well.
Gene Stoner, in a blank-sheet-of-paper design, looked at smaller, high-velocity .22 varmint and target rounds. At the range of 300m, the .222 Remington could still penetrate a GI steel helmet, which was one of the criteria. He redesigned the round a bit, and the .223 Remington was born.
The smaller, lighter bullet meant a flatter trajectory, lower recoil, and less cost. It also meant that the total system weight of rifle, magazines, and ammo could be made lighter.
Some of the current "faults" of the 5.56mm round result from the rifle-and-ammo combination being an over-achiever, and doing more than it was intended. With a good scope, a good infantryman can hit targets out to 600m. Accurized M16s/AR15s with match ammo are used at 1000 yard matches, and are beating the M14-type rifles.
The "problem" is the M16/5.556 combo is inherently accurate, allowing targets to be engaged far beyond the original intent of the specs. But the lightweight (even the current M885 ammo) bullet lacks quite a bit of lethality at that range. You can reliably hit, but the target does not reliably die at those longer ranges.
There are classified programs to develop a round midway between the 5.66 and the 7.62 in size and weight to address this specialized need. Not-for-attribution comments say the results are fantastic.
Even without a different weapon (some of the prototypes are based on M16 components), it would introduce another caliber of ammo into the supply system. The 5.56mm round cut back on the demand for 7.62mm NATO, but did not eliminate it, since it's still in demand in the machine gun role.
The US now has a 40+ year history for the M16 and its ammo. The M16 series has become the longest-serving rifle in our history. Other western nations have 20 or more years invested in their 5.56mm systems. Not one country has said, "the 5.56mm is crap, we're going back to 7.62".
Oh, and with all those M1913 rails demanded on all current weapons, here's something you could do to your M16 (I think), but not recommended:
