Sounds more like making use of weapons that were in dire need and ammunition at hand. Weapons don't last very long in those conditions, testimony from men that were there show destruction of weapons in short order. They have to be replaced somehow.
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Yes, that's exactly what they were doing and of course it was done in a thorough and organized manner; it's the German army isn't it?
We know from the photographic evidence that use was made of captured Vickers and Lewis guns; that may have been purely local at times. In fact even if there was a policy of collecting such weapons centrally, we know that soldiers in many armies often retain captured weapons for their own use, if their immediate superiors overlook or permit it.
The SMLE probably held little attraction for the average German soldier; they would naturally tend to believe the Mauser was superior. A Maxim-type MG or Lewis gun would be much more attractive.
Obviously rifles would not have been issued to formations until they had been cleaned and inspected and the troops trained in their use. That doesn't happen on the way to the front.
There are WW1 accounts that make clear the Germans' appreciation of the Lewis. It was even known on occasions during informal truces for not only the wounded to be carried back from No Man's land by German stretcher bearers!
In WW1 the Germans most certainly used captured machine guns and were particularly fond of the Lewis.
At first just used as another machine gun, but rapidly adopted for use in "assault teams" and as portable support for trench raiding and "rapid reaction forces". There were apparently enough savvy armourers and "engineering" types to keep a small fleet of them going as long as ammo could be "obtained". See also their use of captured Vickers as "supplements" for the MG08 / 08-15 fleet. Anyone have any info on what the Germans did with the substantial number of Russian Maxims acquired in the Eastern Front in WW1? Changing out the barrel, breech block and a few parts in the feed mechanism on a gun generally the same dimensionally (and often made in the same factories), would seem like a feasible project.
Long-term, reliable ammo supply would be a deciding factor, especially once warfare became much more "mobile" from late WW1 onward to the mass (ish) mechanization of WW2,
Thought I might take a shot at finding the video and it was the first one I clicked on. Start at 0:58
Drei Lilien | German army WW1 footage in Color - YouTube
Just noticed there is currently a WWI German captured SMLE for auction on eGun: https://www.egun.de/market/item.php?id=12811955
There is a documentary film made shortly after the war re-enacting battles of the 1914 campaign, of which there was very little genuine footage due to the all-pervading spy mania, with British troops providing the 'Germans' and naturally using their own smallarms for the purpose. Pieces of this can get cut together with other stuff, and generations of writers just copy the attribution.