My priorities for underpants are that 1) they don't ride up, 2) there are no crotch lines across the buttocks, and 3) there are no lines whatsoever.
I can't find #1 in ready-to-wear, never mind the other two, so I make my own. Until this year, I made them all out of 90 - 100% cotton knit from a self-drafted pattern. These were cut in one piece with only side seams. For normal cotton dress trousers, there were minimal lines from where the elastic cut across my hips/thighs. So I was happy, even if it does take me two-three hours from first cut to final thread snip per pair. People who say they can make a pair of underpants in less time than they can do a load of laundry must not be using a regular sewing machine or they are very fast cutters and sewers. Or maybe "load of laundry" to them means washing and drying, which would be about 90 minutes, not the 30-minute wash I envision. That's still faster construction than I can manage.
Back to cotton versus nylon. Sometimes I wanted something more sensual than cotton, so I figured out how to move the crotch seam so that is it far enough under the buttocks not to show in normal movement. I also figured how how far forward to extend the cotton crotch so that it serves its purpose. I figured this out by putting on a par of underpants and feeling where I wanted the crotch seams and putting pins in those places and then undertaking trial and error as I attempted to reproduce those seam placements on a flat pattern.
I am confused about why commercial underpants and patterns have one crotch seam in the back with the crotch insert loose at the front. I would think it would make more sense to have the seam in the front and the insert loose at the back. This would eliminate the insert bunching up from gravity pulling it down and would eliminate the crotch seam line showing across the buttocks. Is there a benefit to the standard way of constructing underpants?
Anyway, normally, to reduce bulk, I stitch the elastic to the wrong side of the leg opening and then trim the fabric to neaten the edge. I don't turn the fabric over because that adds bulk. Since this year I used tricot, which is thinner than cotton jersey, I sewed the elastic to the right side, then turned it to the inside. As you can see, I don't worry about trimming any seam allowance that sticks out of the fold. I use a normal zig-zag stitch. The stretch stitch on my machine is a horror to unpick, so I try to keep my life simple by not using it when something else works.
For the waistband, I use any method that hits my fancy at the moment. I don't worry about bulk there. I've drafted the pattern so that the underpant's waist sits low enough that even when I wear long johns under my trousers, none of the waistbands are at the same place. This is more comfortable for me.
The pair on the right below are made with this plush elastic. It is the most bulky of all the elastics I used and I only used it on one pair. It also took up more fabric than any other elastic. The crotch on this pair is an inch narrower than the one on the left, which is finished with fold-over. The crotch width of pairs finished with regular picot elastic are somewhere in between, but closer to the width of the fold-over pair.
I wore this plush elastic pair for a day, including a 30-minute brisk walk, and didn't have any problems, but it is just slightly on the not-quite-comfortable edge. Not enough to be a problem, but enough that I won't again use that elastic for a leg opening.
I'm curious to see how the tricot holds up. With my cotton pairs, the elastic falls apart before the fabric or seams do. I just got rid of pairs I've had for three or four years that finally got too thin and stretched out. I figure if I'm spending three hours making one pair and it lasts me three years of comfortable active wear, even with weekly washing, that's a good value.
Two more thoughts: In the photo below, it looks like the side seams are shifted towards the front. When I wear these, they are at the center side. The back of the underpants is wider than the front because that is how my body is. For the legs, I stretch the elastic more or less evenly all the way around, but maybe slightly more around the back and very little through the crotch.