It's not so much that the military thought the 7.62 was overkill for "stopping power" but rather for distance and weight. An average soldier doesn't accurately engage the enemy at 700yds, and they can carry 2-3x more 5.56 than 7.62.
and that small bullet will stop someone or something in its tracks.
WAY too many factors make this a questionable statement.
That said, the lower recoil of a 5.56 does make rapid follow-up shots easier. Quite a few enemies on the battlefield continued to fight after 1 hit. A lot fewer continued fighting after taking multiple hits.