55 gr. M193 is still used by the entire US military for training since its cheaper.
As others have said choosing 5.56/.223 (but really any cartridge) bullet weight is based on intended use/purpose. As others have said twist rate is a key factor. Heavier (therefore longer) bullets require a faster twist rate to stabilize the bullet until it slows enough down-range to destabilize. But barrel length is also a factor with shorter barrel lengths typically result in lower initial velocities which results in a shorter de-stabilization range.
The US military chose the 62 gr. weight to add range & lethality with the M16A2 model, and standardized the 1:7 twist range on the M855/M855A1/SS109 spec-loads, although the 1:7 twist rate was needed stabilize the longer/heavier NATO L110/M856 5.56×45mm NATO tracer bullet.
The M855/M855A1/SS109 was developed to meet a NATO requirement that the bullet be able to penetrate through one side of a WWII U.S. M1 helmet at 800 meters (which was also the requirement for the 7.62×51mm NATO). The intention was that the 62 gr full metal jacket bullet, with a mild steel tip, moved the center of gravity rearward, increasing flight stability and thereby the chances of striking the target tip-first at longer ranges.
However, there have been terminal performance deficiencies with the M855/M855A1/SS109 during its use in overseas conflicts. These problems have primarily been manifested as inadequate incapacitation of enemy forces despite their being hit multiple times by M855 bullets. These failures appear to be associated with the bullets exiting the body of the enemy soldier without yawing or fragmenting.
Cost is also a actor. I use M193 55 gr-equivalent for 95% of range/training use, and don't se a need to use more expensive M855/SS109 for that purpose. While I have M855/SS109 use it with a commercial 64 gr. load that provides better terminal performance on soft-targets in a mixed self-defense loadout.
I max out with 75 gr loads for LR CMP competition stages.
My .02