@Matt B what I found worked for straightening my wire was to double it over halfway (my strands were only 1m long so results may vary for longer strands) and put the halfway point over the bottom of my foot at the end of my outstretched leg. Then I grabbed each section of wire in each hand and pulled and “slid” my hands up the wire towards my hip until I reached the ends of the wire. One “slide” up the wire was enough to straighten it sufficiently.
This did still take some time to do 15m of wire and I ended up with a blister on the inside bend of my index fingers from the friction.
The wire I'm straightening is in 8 foot sections, and I am not well enough endowed in the outstretched leg department to cover that. The wire is pretty thick, enough that I need to use gloves to not tear my hands up just manually straightening more than a few strands in a row. The only other way I have found is to use a screwdriver as a "hard edge" and run the wire past it the way you would a piece of paper on the edge of a desk. It takes a few passes to get it straight enough for a neat coil.
I dealt with the straightening by using gloves, gripping the blade end of a screwdriver, and running the wire between my fingers, dragging it over the shift of the screwdriver with a pair of lineman pliers. Almost as if straightening a wrinkled paper on the edge of a desk.that brought the time to straighten a 10 foot stand from 10 minutes to 20 seconds.
I finished my little batch of wire, which should last me for a short while... or 20 years. Same thing.