Given the rapid growth in hard armor that can take a few 5.56mm rounds before failing, I wouldn’t be surprised to see militaries shift more towards a 6.5 or 7mm cartridge with a hardened steel penetrator.
What most people don't understand is the troop loadout for combat ops. Men are not dripping with weapons and rounds per Hollywood. The standard loadout is an M4 rifle with 6 30-round magazines (180 rounds). If you are issued a handgun, that is 3 15-round magazines for an additional 45 rounds of ammunition, albeit in the "weak" 9mm, which is a get-off-me gun meant for house-to-house, close quarters fighting where a long gun would be unwieldy.
As you suggest, there is a large push to go to the 6mm/243/6.5, which offers SUBSTANTIAL performance benefits over the 5.56. A .243 (6mm) is pushing more ballistic energy at 500M than a 5.56 does at 100M. That's a lot more. And the rounds don't weigh that much more or take up that much more space. Recoil, likewise, is negligible at best due to the gas-operated platform. But, NATO... NATO is sold on the idea of 5.56 and 9mm. I think that DMs should be issued the better long guns to deal with issues the 5.56 cannot handle.
That might be the reason for continuing to use 5.56mm, but it looks like America actually adopted the cartridge first.
The .30 will reliably put opponents down very well, even with the required FMJ rounds. The .30 stuff is far better at hard barrier penetration than the 5.56. The .30 turns cars and walls into concealment rather than actual cover. The 5.56 suffers here unless it's a crew served weapon that can chew through barriers.
I understand the reasons for going to the 5.56. I lived through the transition from the .45 to the 9MM, which was and is lamented by the USMC. The 9MM is analogous to the 5.56/.30 issue. NATO compatibility is the sole reason, but the troops suffer from the lack of power these rounds offer. In the gas-operated weapons used, recoil difference is negligible, since the system soaks up the recoil pretty well, and when the adrenaline runs, recoil and noise are not really noticed anyway.