Ah, No. The AR10 was a competitor to replace the M1Garand. Competitors included the FN FAL, some British rifle and the Springfield rifle that became the M14. It lost. Badly. Very badly.

The AR15 was a competitor for the DOD requested "light rifle" in .22 caliber. The initial request for proposal (both ammo & rifles) was long before RSMcN had anything to do with the DOD. Neither the Winchester or Armalite rifles were adopted initially. Remington won the ammo competition over Winchester but there was no contract awarded at that time.

Curtis LeMay actually got interested in the AR15 as a firearm for his airbase defense teams. RSMcN jumped on the bandwagon because a rifle is a rifle, just like an airplane is an airplane and a truck a truck. Regardless of the possibly differing needs of the end user.