Years ago I saw an infomercial for this handy plastic tool to thread elastic through casings in "mere seconds". I thought it looked neat but had no need for it at the time. Fast forward to present, I have spent more time than I care to think about trying to feed a safety pin through yards of casing. All the while thinking to myself of this plastic tool that I can't seem to find anywhere.
I finally have found a better way than the safety pin - a zip tie. It is cheap and essentially the same as the plastic tool I saw once upon a time ago. Simply create a loop with the zip tie, attach your elastic either with a small safety pin or by tying the elastic onto the zip tie, then feed it through the casing.
Zip ties are available in any number of thicknesses and lengths as well. I use a tiny one for threading elastic through baby pants or a large one for threading shopping cart covers. So simple that I'm kicking myself for not having thought of it sooner!