When I first replaced my handlebars years ago, I found feeding the wires through them to be a nightmare. I had silicone spray on them, I was using a feeder taped to the end of the wires, I was pushing, I was pulling. Probably took near an hour to get each side in.
After rebuilding my controls last year I pondered this problem as I waited for the weather to warm up to put them back on. I wanted something that had the rigidity to feed through the handlebar tube, and with an end that could loop around the bundle of wires to be pulled through. I knew just such a thing: a guitar string, low E or A! Has a convenient little brass loop on the end that you can feed the string through, so the more you pull the tighter the loop becomes. Whole thing took about 3 minutes this time.
Pic attached. Sorry I didn't take another one with it wrapped around the bundle of wires, but I'm sure you get the idea.