Originally it was designed as the model 62 in 7.62mm NATO. When it was recognized that 5.56mm was the way of the future, it was redesigned into the 63.
The entire weapons system was tested extensively by the Army and USMC. The Army tested the rifle and carbine as the XM22 and XM23, and the LMG as the XM207. During these tests, problems showed up and the 63A came out to correct some of them.
The Navy tested the LMG and used a short, fluted barrelled model as the MK23 for the SEALs.
The USMC was pretty hot for the system. It would solve alot of logistical problems with Marines afloat, since the number of unique spare parts could drastically be reduced, and parts from one gun could be used to repair another, even if it might not be the same version. Also it enhanced training in that you trained with one type of weapon, and you knew all of them because it was a system with few uniqe parts. They managed to get the test platoon mentioned above going, but even though it showed that it could be an attractive system, they never got funding to proceed any farther and it was dropped.
You have to remember what was going on at the time and place it in that context to understand why it was left unfunded. Having a different weapons system for the Marine Corps was opposite of the direction that McNamara was taking the DoD. Remember him forcing the USAF to buy F-4 Phantoms and A-7 Corsairs? He asked some pretty logical questions as to why services needed different colored shoes, to why they needed this that or the other thing that was uniqe to the service. In many instances there really wasn't any good reason that one service needed, say different socks, than another. The result was a great deal of financial savings on buying just one item and using it in all services, rather than buying a different, but similar item for each service. As with the F-4, this practice was taken as far as it could go. So The logical question was asked, "Why are we going to buy a different 5.56mm rifle when we already have one?" Hey, alot of people asked the same question about the HK XM-8 right here on the board.
With the M16 program mature to the point of millions of dollars spent, buying the 63 would run the risk that it too would probably need some development as well, so you're re-buying that process with a different weapon. Remember, McNamara came from FORD, and the American auto industry is definately one that feels you should use a design as long as possible, even if it's out dated, as long as it's profitable. The DoD had already paid to fix the AR-15. It had no intention of paying to fix yet another weapon.
Also you had a great deal of "old school". Many senior officers felt that 7.62mm was what Marines should be shooting in the first place, not 5.56mm. So ANY 5.56mm wasn't going to get much backing from dinosaurs. Those that could see ahead liked the Stoner 63, but even today there's people (some right here on this board) that opine the replacement of anything they used "back in the day" with something new.
Finally, we were in a shooting war at the time, and the Army was the lead service in the fight (similar to today) due to numbers and money. Once you got above the division level, it was pretty much an Army show as far as logistics and even command went back then. So the DoD felt it would be simpler to have everyone equipped the same in theater. This does make some sense, as in order to support a uniqe rifle/LMG for the Marines, you'd either have to run a parallel support system, or add those lines of supply into the existing Army system, which would have no other purpose than support Marine units there. Either way would cost money, and probably off-set any savings that the weapon system achieved.
Money, politics, timing, whatever. It's usually the same story. The weapon itself was a good one, and it would have been interesting to see where it would be today.