The 5.56mm was a product of its times. It may not be here to stay but it is certainly here. Most larger armies that are likely to be deployed beyond their own borders use it now, along with light machine guns (whatever they're designated--squad automatics) in the same caliber, if they don't use an AK based system, and some of them are even in 5.56mm.
Since the introduction of the M1 rifle, nearly every sort of combination of rifle, submachine gun and automatic weapon has been used at rifle company level. I was going to start with the MP44/Stg44 but I decided it made more sense to go back a few years and start with the M1. It may not have been revolutionary but it was certainly the first successful semi-automatic infantry weapon (aside from pistol caliber weapons) to be fielded in large numbers. Others were introduced during WWII but were not generally as successful for various reasons and none were produced in the same numbers. The M1 actually continued in use in the National Guard well after the M16 was adopted and it was widely used in other armies.
Other armies generally replaced their bolt action rifles with semi-automatic rifles between the end of the war and sometime in the 1960s. There were even new bolt action rifles being manufactured for military use in the 1950s, though I don't think many of them, if any, saw wartime service. And of course India was manufacturing the SMLE in 7.62 NATO even after I got out of the army. The full power infantry rifle is still around in some places, of course, though I'm not sure many new ones are being manufactured, even with plastic stocks. The medium machine gun, however, is still around.
The real variation comes in with the squad automatic. Some had a dedicated squad automatic in the same caliber as the infantry rifle, while others had a regular light machine gun somewhere in the company. Many tried using a light machine gun at squad level, including the US, in some units. The British used the MAG at squad/section level but just as often still retained the Bren, the chief difference being box magazine versus belt feed. But you don't read much about machine guns here, understandbly I suppose.
Even during WWII, some armies had what were essentially designated marksmen. A sniper had more powerful optical equipment. That's the sort of thing that apparently has to be rediscovered every time there's a shooting war. These days the designated marksman concept is a working thing and appropriate equipment is liberally distributed within the army, even to units that really wouldn't make much use of them, like tank units. So they just pass it on. Chances are, the unit commander, typically the company commander or maybe the battalion commander have to buy into the concept or they won't be used properly.
The designated marksman is more than just equipment. He goes on a course and, hopefully, he is a "select man," as the expression was back when a rifle was something special. It only makes good sense to start with someone with promise. In the meantime, everyone else has fancy sighting equipment anyway. Since there is a regular automatic weapon and a grenade launcher, not to mention a medium machine gun somewhere not far away, an infantry unit is well equipped to deal with a variety of battlefield situations with what they already have.
From everything I hear, the 5.56mm is well suited, if not perfect, for the role it is intended to fill. It would be interesting to hear Stoner's ideas on the project from 50 years ago that started us on the road to where we are now with our infantry weapons. Armalite, you know, came out with other weapons that mostly didn't catch on but still all saw some active service, yet we continue with basically the original weapon from before I went to basic to learn to shoot an M14. Both the rifle and the cartridge are bound to be replaced someday but with what, who knows?