The fundamental purpose of full-auto fire is to increase the probability of hits on moving targets or at extended ranges.
The standard M14 service rifle proved to be unsuitable for full-auto use. Standard barrels overheated very quickly causing the rifle to become unusable. Recoil of the 7.62X51 made control very difficult in full-auto mode, even when used with the standard detachable bipod.
Every standard M14 I used in the Army had the selector switch removed, allowing only semi-auto mode. Focus of all training was controlled and accurate firing, as well as conserving ammunition in combat (7.62X51 ammo is HEAVY and the magazines are bulky, so each soldier can carry only so much; typically 5 magazines of 20 rounds each).
Basic training (also at Ft. Jackson, SC) in 1968. My rifle was manufactured by TRW (satellite company in California), very heavily worn from countless cycles of trainees constantly disassembling, reassembling, humping, and rifle training. When held in one hand and shaken it sounded like a steel bucket full of nickels. But on the rifle range it performed perfectly every time. Qualification course was 82 pop-up silhouette targets at ranges from 75 to 360 meters, timed exposure of the targets depended on range (most in the 5 to 10 seconds range, with a target presented and not engaged counting as a miss), with 84 rounds allowed. My rifle produced a perfect score, providing me with the Expert Marksman badge, a 3-day pass, and the post rifle trophy for our Day Room display (along with huge bragging rights for my Drill Sergeant).
Later, during infantry AIT, we trained on the M14E2, specifically designed for use as an automatic rifle. Heavier barrel, full pistol grip stock, built-in steel bipod, modified flash suppressor (about 15 lbs IIRC). Always used from supported positions (foxhole, breastwork, prone). Emphasis was on controlled 3-round bursts, requiring concentration on the trigger. Training targets included multiple man-size silhouettes, old vehicles, various suspended moving targets, all primarily at extended ranges (300 to 500 meters). Qualified with the M14E2, then never saw another one in actual issue or use; all of the infantry units I served in or observed had the M60 light machineguns by that time so the M14E2 was pretty much superfluous.
Got to Vietnam in 1969. By that time much of the turmoil of the early M16's had been experienced and the M16A1 was pretty much standardized. There were still a few M14's on hand, usually reserved for senior NCO's who had grown up with M1's and had a lot of experience with the M14's. I was eventually able to trade for one from a ARVN source known to our Vietnamese interpreter, and when I left I had no difficulty finding a new home for that rifle.
For shooting (especially at longer ranges) or for static defense I would choose the M14 any day. For patrol or fluid combat situations my nod goes to the M16A1.