Hi - I don’t know spit about .30-06, especially compared to Chris Punnett, whose knowledge on anything and everything is tough to beat. But, having served in the Army and trained on oversized, cut-away models of the BAR, .30 Carbine, M1 Garand .30 Rifle and the M1919A4 and M1919A6 machine guns, I can tell you with certainty that your oversized .30 Model of 1906 cartridge was for demonstrations (to use your perfectly valid word) in the classroom of the inner workings of the various infantry small arms in use at the time.
These double-sized models allowed a moderate class of 30 to forty soldiers to see how their various weapons operated. The plastic cartridges used came in OD like yours, and then also in yellow and black, and perhaps red, in plastic cartridges a little better proportioned to the original shapes than were the OD ones. There was a .30 Carbine caliber version also, which I have only seen in OD but perhaps was made in other calibers as well.
The .30 Model of 1906 versions came loose for packing in the oversized BAR magazine, insertion in oversized cloth belts of the same pattern as the normal whitye cloth belts used, along with steel-link belts as well (but no double-size one of which I am aware), in the actual Browning machine guns. There was also a rather simplified and cheaply made sheet metal (not blued, at least not the ones I saw) 8-shot clip for the M1 Garand instructional model.
Not much else I can say. I used to own one of the Browning .30 MG models, but gave it to a friend who had more room in his gun shop to display it than I did in my den.
I hope this offers you some insight into the use of these interesting “demon-