I haven't seen that moview, and I am not a historian, but just with common sense and my little history knowledge, I'm pretty sure that's all about the fact that a dead cow is not a fridge, as someone noted in the comments.
Lead? It was a war time, and not the present times. And it was not the "present kind of war" where you've got lawyers waiting for the end of it to begin a class action lawsuit for every single thing that went wrong or not according to the book.. I'm pretty sure that if the army were in pinch and low on supplies, they'd make use of any cattle, pigs, sheeps, horses, whatever, and regarding bullets, yeah, they'd probably not used them, but not due to the fear of lead, but to save the limited supply of bullets for actual combat.
Dragging a living stock along with troops always means slower movement comparing to prepared food (or use more motorized transport, expensive), and also means delegating people to take care of it. Still, that's a very good option, especially if there are any problems with food. A large herd could be held for as long as there's some grass/etc for them around, and even for some time without feeding them, and then they can be used at rate fitting the needs. If a herd was killed all at once, if the meat already started to spoil, it's useless. If they were lucky to find them still fresh, some of that could be used immediately. But unprocessed meat gets spoiled quickly, you can't just keep draggin a dead cow for a day or two and eat it later. You have to process them. If the herd was large, it would be very hard to process all them, not every soldier knows how to do it, and it would be even harder to preserve it for later use.
I've found an article describing some aspects of food supply at WW1:
This article mentions rationed dry food, tin cans, and introduction of field bakeries. No living stock. Nothing about freezing. If I recall well, typical contemporary freezers were invented somewhere around 1900. They could be already in use in long term camps, but I think they'd be useless for mobile troops. Freezing requires power, or fuel, and I think both were at premium. If I'm correct at this, this means that for preserving such fresh meat, they'd have salting (needs salt or chemicals), drying or smoking (needs time, fire, smoking wood, and at least makeshift construction work), or cooking/boiling/roasting/etckitchenworks and putting that in jars/etc. Each gives different period of preservation, each needing some skills at least to start the process (but immediately, can't wait a few days for someone to arrive), and also they'd need some proper containers, relatively clean - so i.e. leftover fuel barrels might not do the trick, and so on.
I'm highly skeptical they could make use of that resources, unless they were well prepared upfront for that for some reason.