The .50 BMG was originally intended to be an anti-materiel round. It does have excellent accuracy from some platforms like the MCM Tac-50, but most .50 weapons systems are designed for anti-materiel applications with corresponding accuracy.
The .338 LM was designed dual-purpose- light anti-material and longer range sniping beyond the capability of the .308 NATO. The project was started by RAI with the goal of pushing a 250 grain bullet to 3000 fps...and they used a necked-down .416 Rigby case to do it. The brass production was unable to handle the pressures, and they target velocities were not obtained and the project was abandoned.
Sako, AI, and Lapua eventually picked the project back up and the rest is history...
So to the OP, far as "why not the .50 BMG" the answer would be many fold... there is such a thing as overkill, and if you don't need the expense of the weapon platform, the expense of the ammo, nor the kinetic energy- then you don't use the .50.
Big difference between a 250 grain projo at 3000 fps- and a 700 grain at the same velocity.
It's no different than any other application-based selection of caliber and weapon platform, right?