.270 is necked down from the .30-06, IIRC.
The whole principle of the intermediate cartridge... research was in progress before the Second World War, but that's where it really came to fruition. German research showed that firefights most often occurred at a range of less than 200 meters. Topping it off, the standard infantry weapon of the German military was the bolt action Mauser, and the submachinegun typically filled the role possessed today by light machine guns and dedicated automatic rifles. And, it was found to be a combination of too much/too little on both counts. The submachinegun could put out a lot of lead in a hurry, but a pistol cartridge was hardly ideal for many combat scenarios. The K98 had power, but a slow rate of fire, and was comparatively unwieldy.
The most famous (though not only) solution in finding a compromise between the two came in the StG43, which fired a shorted 7.92x33 round - sufficient for the typical firefight, powerful enough to get the job done. reasonably manageable in automatic fire. It didn't have all the pros or cons of either weapons system it augmented, but possessed some of the best features of both. The Soviet Union were the first to actually made widespread use of this principle, although the British were the ones to take it to the next level with the EM2. In fact, the FAL was originally designed to fire the intermediate cartridge designed for the EM2. However, US pressure forced NATO to adopt the 7.62x51 cartridge, and that was the end of the EM2.
The 5.56 was a further adaptation of the same principle.. lighter weight made for more manageable recoil in automatic fire, it allowed soldiers to carry more ammo, etc. Ironically, the US essentially pushed NATO into adopting the 5.56 after essentially forcing the concept off the table in the 1950s.
Interestingly enough, evaluation of US actions in Vietnam seemed enough to convinced the Soviet Union, and they developed a smaller 5.45x39 round for the AKM's replacement.
The 5.56 is not perfect. No round is. It does what it's supposed to, same as any other round. A Combat Basic Load for a US soldier is 210 rounds - even as a line medic, I carried this. A typical basic load for a force using FAL, G3, or other 7.62x51 rifles was 100 rounds.
So, it's all a bit of give and take.
and although almost every country uses a different gun, they all take the same STANAG magazines
This is not entirely accurate. STANAG 4179 was never actually passed, and while a great many NATO countries
did adopt the M16 magazine, there's still a great number who did not. Germany does not use the STANAG magazine. Spain is replacing their STANAG compliant CETME Model L rifles with the non-STANAG G36. Poland is using an AK variant, which does not use STANAG magazines, etc.
The Warsaw Pact was better standardized on small arms magazines than NATO ever was, with almost every country using the AKM (Czechoslovakia being the exception to the rule), whereas, at the time, NATO had no standardization requirements (and still really doesn't) on small arms magazines... some NATO countries used the G3, Spain used the CETME, some countries used the FAL, but the British variations of the FAL used a different magazine, and so forth.