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Having handled both 7.62 (G3) and 5.56 (HK416) in my army time, I can tell you a 7.62 does not work as you seem to be suggesting.
Assault rifles, even ones as accurate and light as 416’s, are probably not going to hit a person on the first hit. More than three fourths of bullets fired by Norwegians in Afghanistan (and this was our most well-trained regular forces) were estimated to have missed their target. At that point having a bigger bullet would be irrelevant, since you wouldn’t bring enough with you to saturate the target.
You can’t really train troops to be as accurate as you’re suggesting with assault rifles, which as I said are not effective at “picking off targets.” For that we have designated marksmen (who in their role do actually use 7.62).
Carry weight becomes a factor. With the G3 we would lug 4×20 bullets around. With the HK416 it was 6×30. Already the 7.62 is at a massive disadvantage since volume of fire, a big deal in infantry doctrine, is more than halved, since because 7.62 weighs more than twice as much (28g vs 12g) you have less of them in fewer magazines.
You’re right in that you’ll spend less time engaging a target, but that’s because you’re out of bullets.
Accuracy and weight notwithstanding, the power of the 7.62 quickly becomes a liability. Firstly there’s the issue of overpenetration, where a bullet just flies through the body without tumbling much. Secondly, in a world where urban warfare is looking more and more normal, a bullet that penetrates as well as a 7.62 is a big danger if there are civilians hiding anywhere near your opponent.
Essentially, until most armed forces are kitted out in Batman armor, the 5.56’s strengths more than outweigh its weakness.
Edit: I base my comments on interpreting “fewer bullets, less time engaging a target” as saying you’d spend less time bringing someone down.