What was the benefit/reason for fielding this new rifle (T20) when the BAR was already widely fielded? I'm clearly ignorant on this phase of small arms development!
In short: 20-round box magazine (T-20) vice an 8-round clip (M1) and 11 lbs and change (T-20) vice nearly 20 lbs (BAR). The full-auto feature was controversial and caused a lot of problems when trying to develope the rifle. It was found to be too light to fire accurately on full auto even with the recoil-break / climb compensator. However, there were several "product improvements" to the M1 Garand which had been deferred in order to not disrupt production. (The roller bolt was a big one as was a box magazine, but there were others.) So, the decision was made to incorporate all of the recommended changes into a new design and field it for the anticipated invasion of Japan.
The T-20 was never, however, envisioned as a replacement for the BAR. It was intended as a "product improved" replacement / supplement for the M1 Garand. The full-auto feature was intended only for occasional use when an extra volumn of fire was needed. It wasn't supposed to be a new squad automatic weapon.
In any event, the invasion never took place, and the new rifle was shelved with only a couple hundred initial production rifles made. With millions of M1's already on-hand and the war over, there was no need for a new rifle. The continued manufacture of M1's during the '50's had more to do with the "military-industrial complex" spreading its footprint around the country to limit damage in the event of a nuclear war than any real need for new rifles.
The really interesting part of the story, IMO, is the development of the M14 which took almost 14 years. Very few people are aware anymore that the US military came exceedingly close to adopting the FN/FAL. (So close, in fact, that the Infantry School produced a Field Manual for that rifle too.) The proto-Stoner AR was also a contender, but really suffered from too many early teething problems at that time ('50's). In the end, the M14, really just the ultimate "product improvement" of the M1, was clearly the better rifle, but it was too late. Defense Secretary McNamera had the a** at Springfield Arms over the whole waste of time and money in fielding the new rifle (the M14). He's the one who pushed the M16 on to the military and canned the M14 (and closed Springfield Armory, too, BTW).
At any rate, there's more to the story than meets the eye.