Why does my elastic keep rolling inside the waistband I sewed?

Yeah, the corkscrew thing is almost always a casing width problem. If the casing is much wider than the elastic, the elastic has room to fold over on itself and once it starts rolling it just keeps going. A 40mm elastic wants a casing maybe a few mm wider than the elastic, no more. If yours is like 15mm of slack in there, that's your culprit. The wide elastic advice you read is still good, wider ones do resist curling better than narrow ones, but only if they're snug in the channel.

And yes, anchoring the elastic at the side seams is the fix people use for exactly this. Overlap the elastic ends, zigzag them togehter into a loop, then stitch thru the whole sandwich at each side seam when you close the casing. Some people also do a vertical stitch through the elastic at the centre back or centre front to keep it from migrating, though with pleats at the front you might not want a visible stitch line there. Anchoring at the side seams alone ussually stops the rolling. Next pair, consider stitching the elastic directly to the waistline seam allowance in quarters and then folding the casing over it, that way it physically can't twist.

One other thing worth checking: some elastics are just soft and stretchy and roll no matter what. The firmer woven type holds its shape alot better in a wide waistband, the soft knit kind is better suited to cuffs and necklines.

For the eyelets, yeah, interfacing behind them is the trick. A little square of fusible interfacing behind each one, or even a double layer of the twill itself as a stay, stops the fraying and gives the eyelet barrel something solid to grip. In a heavy cotton twill you also want the hole punched or cut clean rather than snipped with scissors, a ragged hole is what frays out around the setting. Hammer-set eyelets with a washer backing hold up way better than the prong type in thick fabric too.

For the pair you've alredy made, you don't have to bin it. I unpicked about 5cm of the casing stitching at one hip, pulled the old elastic out, threaded a piece of firmer woven elastic through with a safety pin, overlapped the ends and zigzagged them, then stitched through the side seam to hold it. Took twenty minutes and that waistband has stayed flat since. Also on the drawcord, I stopped bothering with eyelets on heavy fabric and just sewed two buttonholes on the inside of the waistband instead, cord comes out between the layers and looks cleaner from the outside anyway.

Not sold on the claim that anchoring at the side seams on its own stops the rolling. I did exactly that on a pair of shorts last summer, stitched through the whole sandwich at both hips, and the elastic still corkscrewed within a few wears. Turns out mine was twisting between the anchor points, not at them. The pleats are the bit I'd look at here too, they change how the front of the waistband sits compared to the back, so the tension is never even and the elastic gets pulled out of flat. I ended up adding those vertical stitches at the centre back and it finally behaved.

One thing I'd add on the elastic itself is that a lot of it relaxes permanently after the first wash, so if you cut it to her actual waist measurement before washing the finished pants, you can end up with slack that makes the rolling worse beacuse a slack band has nothing keeping it flat against the body. I now pre-stretch mine a few times along its length before cutting, and since doing that I've had far fewer waistbands go lumpy on me. Also worth measuring the joined elastic while it's lying flat and relaxed rather than stretched, because if you overlap the ends by too much the finished circumference comes up short and the excess fullness just folds in on itself in the casing.

Update: took the unpick-and-swap advice and opened up the hip seam last night. Turns out my casing is easily twice the width of the elastic, so that explains a lot. Restitched a snugger channel and it's been sitting flat through a whole day of wear so far. Still no word on the eyelet fraying though, might just try a scrap of fusible behind the next set and see.