This film is about giving soldiers confidence in their equipment, a mild form of propaganda but it's not an entirely inaccurate assessment either.
It certainly would not have gotten anyone killed.
I'm not sure how that would have gotten an increased amount of soldiers killed? Any machine gun is deadly either way. Any solder with an IQ of 50 and above would understand that, and the information in the video. The fact that it's accurate information and is correct only helps better understand the practical side of weaponry use and functionality. It's probably a better thing to inspire some confidence to any degree as opposed to telling them something like, "best of luck because the germans have better guns that shoot faster". There was a video I watched a while ago comparing the practical shooting capability of I believe an mp40 to a modern HK ump or something and because of the slower cyclic rate of the mp40, it outperformed the ump in controllability and accuracy. I'm sure there are a LOT more where that came from too.
In terms of small arms each of the beligeriants had an area that they held the advantage.
Our troops entered the war with the best infantry rifle by far, and we also had the unique M1 Carbine, the worlds first PDW. I think we also had the best pistol but that doesn't matter.
The Germans had the best, and first medium or general purpose machine gun and an excellent submachine gun. The Brits had the best light machine gun and the Russians had the best submachine gun and some very good machine guns.
As the war went on the Germans developed the best infantry small arm of the war in the STG 44, not that it helped them much.
Also of note we were the only ones to field heavy caliber machine guns so widely.
Just as our troops dreaded the buzzsaw wail of the MG42, the German troops were equally terrified of the heavy pounding of Ma Duece.
The Italians were pretty much straight sucking in small arms, as they were for every other aspect of military equipment, though they had a good submachine gun also.
As for the Japanese........eh. They also had an excellent light machine gun and their heavy machinegun was far better than was recognized at the time. The Japanese did have a variety of lightweight and highly portable mortars at their disposal.