I was confused at first when you asked if the 8-flow is relevant, when like, the 8-flow is a key input that the cycle double cover is built out of. Then I realized, oh, I guess technically they're not using the 8-flow, they're using the Z_2^3-flow. But like. The existence of an 8-flow and the existence of a Z_2^3-flow are equivalent, and I gather most graph theorists are going to talk about it in terms of the existence of an 8-flow, so noting that having a Z_2^3-flow is equivalent to having an 8-flow helps the reader to put this information in context.

I'm not sure why you find this proof so hard to read. I found it mostly quite readable (and the definition of L is straightforward? I wouldn't have written it quite that way but it's hardly inscrutable), although I feel like some parts are maybe lacking some exposition to explain the reason for certain things -- it doesn't feel written "in order". I also don't like that it's not cleanly separated into theorems and proofs -- some of the proof occurs in parts that aren't set off, for instance, and there isn't even a proper main theorem statement! But overall I was able to get through it without a lot of trouble and I'm not even a graph theorist...