The way I have done it is to tape the wire bundle together at one end and start feeding the braid over it.
@Greg's idea works too.
Push the braid onto the end so it expands and moves onto the wire bundle.
Keep working it along the length of the bundle until you get it where you want it.
It's a little hard to explain but this video shows the process pretty well- https://www.youtube.com/watch?v=xWR4yYGd3u4
Check it out from about 0:40 onward.
Unless you heat-seal the ends of the braid as you cut it, it is going to fray (and maybe even then).
There isn't much you can do about that other than to be sure to cut the braid maybe an inch too long on each end and trim off the frayed ends before you apply tape or heat shrink over them.
I have several rolls of braided wire covering in differing diameters in my workshop and use the stuff all the time (mostly on cars).
It's not that bad to work with once you know the trick.
As for getting it into the e-chain, run a single wire through the chain, tape the end of it to the taped end of the wire bundle, then pull the wire through the chain working the bundle along with it.
This is like snaking wires through conduit or walls for home or office wiring.
I share your frustrations on this kit, believe me, but other printer kits are as bad or worse for their own reasons.
And yes, you have to take off the connector shells when doing this. Mostly the stepper one.