The smaller bullet
is the point. A 7.62mm bullet is overkill if you're aiming at a person less than half a kilometer away. The essence of the assault rifle concept is to throw away that excess individual bullet power in order to gain
volume of fire.
The magazine size alone is not a good indicator of practical rate of fire; that has more to do with recoil, which the M16 reduces by using a smaller bullet.
For an understanding of the assault rifle concept, I recommend reading the history of the first assault rifle, the Sturmgewehr 44 (https://en.wikipedia.org/wiki/StG_44#History):
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In the late 19th century, small-arms cartridges had become able to fire accurately at long distances. Jacketed bullets propelled by smokeless powder were lethal out to 2,000 metres (2,200 yd). This was beyond the range a shooter could engage a target with open sights, as at that range a man-sized target would be completely blocked by the front sight blade. Only units of riflemen firing by volley could hit grouped targets at those ranges. That fighting style was taken over by the widespread introduction of machine guns, which made use of these powerful cartridges to suppress the enemy at long range. Rifles remained the primary infantry weapon, but in some forces were seen as a secondary or support weapon, backing up the machine guns.
This left a large gap in performance; the rifle was not effective at the ranges it could theoretically reach while being much larger and more powerful than needed for close combat.
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In the spring of 1918, Hauptmann (Captain) Piderit, part of the Gewehrprüfungskommission (Small Arms Proofing Committee) of the German General Staff in Berlin, submitted a paper arguing for the introduction of an intermediate round in the German Army with a suitable firearm. He pointed out that firefights rarely took place beyond 800 metres (870 yd), about half the 2 km (1.2 mi) sight line range of the 7.92×57mm round from a Mauser Gewehr 98 rifle or less for MG 08 machine gun. A smaller, shorter, and less powerful round would save materials, allow soldiers to carry more ammunition, and increase firepower. Less recoil would allow semi-automatic or even fully automatic select-fire rifles, although in his paper he called it a Maschinenpistole (submachine gun).