From what I remember seeing at a fun-show one time, the 20rd mags for the M1903s were not detachable mags like the BAR mags, they replaced the floorplate on the milled triggergaurds. They came with another spring and used the '03 follower, they didn't have feedlips, so you couldn't load 'em untill they were installed on the rifle.
I've heard a few theories of how these were used/issued.
1- Guys in front line trenches would use rifles fitted with this mag while on sentry duty, and trade off with the next guy. This is so the sentries could lay down continuous fire untill everyone else could get to the parapet in the event of an attack.
2- Aircrewmen would keep these rifles( and shotguns ) in the plane for offense/defense, before mounted machineguns came about
3- Sgts and squad leaders would carry them, so they could spend more time and concentration directing their men, and less time loading their rifles.
These all make sense, but who knows what the real reason was, maybe just a solution to alot of different problems.