Cotton works fine. I do have the blend underwear too, but $18 for a pair of underwear isn't necessary.
I've been wearing cotton underwear for 40 years and never once had a chaffing issue (and I exercise a LOT). Maybe you're getting the wrong size?
Eh, I buy that kind of expensive exercise underwear, but reserve use for strictly exercise.
I have, however, found that the Hanes X-temp underwear (both cotton blend version and 100% poly) are freaking phenomenal as everyday wear. I've sworn to them 100% now that I've got a week's worth+ of pairs, at least when it's not cold, we'll see about winter when that rears its ugly head again. You get 3 or 4 pairs for that $15 dollar mark.
A big, BIG tip for people who do get any kind of performance blend, even if it is 100% cotton but designed to wick moisture (some X-Temp is like that, and there are others like UA's Charged Cotton series)... never, ever, ever use fabric softener.
Ever.
I've committed to a regimen where all my "performance apparel" is washed separately with Sport Wash detergent and dried without dryer sheets (and for those who use it, no liquid softener). Softener completely ruins the capability of the apparel, and I suspect any review on Amazon that said any performance underwear was warm had probably washed everything with the regular laundry. Fastest way to waste money. It may or may not be possible to "save" apparel that has been tainted with softener - I suspect enough washes may revive it but I don't know.