This isn't something that can be done quickly or easily. Simply selecting a new service weapon will take a year or more as it has to be subject to extensive testing to make sure it could hold up in field conditions. There will be tweaks or adjustments the army will want which will require new prototypes followed by more extensive testing. Throw in politics on something this large of a contract and this becomes even more complicated.

So say you've somehow managed to accomplish this mammoth task and selected a new service rifle capable of full auto fire in a breathtaking couple of months. It's the first half of 1951 and US troops are still fighting in Korea. Now comes the next roadblock: you need hundreds of thousands of these rifles to equip the divisions in combat right now. Its unlikely the winning supplier is carrying that kind of stockpile as it would be economic suicide to do so without a contract in place. So even if they ramp up to start cranking them out its going to take months to produce enough to outfit entire divisions. Which brings us to the next major hurdle...

You now need to retrain the soldiers to use a brand new gun. Soldiers that are in a combat zone. And it's more than just how to fire and clean the gun - this is a fully automatic weapon so they need training to not just fire off the entire magazine. They'll need training to learn how it shoots, how it shoots in automatic fire and learn the discipline to not just hold the trigger down. That will take time to do and would be ridiculously difficult in a war zone.

But suppose you somehow got the it produced in number and the soldiers trained to use it. You now have a logistic nightmare in that all the ammo projections are shot. Soldiers can now shoot more bullets meaning your existing stockpiles will be insufficient and you need tens of thousands of more bullets than before and they need to get distributed to the individual units.

Finally a .30-06 cartridge is really big and not really amenable to automatic fire except from bipods. Handing out fully auto .30-06 rifles isn't going to do much if most of the fire is inaccurate. There's a reason that most nations have moved to an intermediate cartridge to make it more controllable in full auto.

Finally the BAR, from everything I've read (and I'm not an expert in this area), wasn't a very good weapon. It's ridiculously heavy - the M1918 weighed around 19lbs. Plus the automatic rifleman had an assistant to carry additional ammo - if everyone had a BAR and you need them to carry their own combat load then the soldier is going to be overloaded and thus move sluggishly on the battlefield and tire out very quickly. Or if they've got a more 'normal' combat weight then they aren't carrying enough ammo.

The FN FAL which would be introduced in 1953 in contrast fired a rifle calibre bullet but only weighed 9 lbs. Britain even opted for a semi-auto only version of the FAL and they had experience in Korea facing the Chinese as well. A contemporary of the BAR was the Bren Gun and it weighed about the same but used 30rnd box magazines and had a replaceable barrel.