While I own and wear pinpoint, etc. shirts, I often find myself reaching for shirts made of 2-ply 100s or 120s broadcloth (or poplin) because it has a smoother hand, is slightly less bulky under a jacket, is marginally cooler, tends to resist being frayed or pilled by my scratchy beard stubble a bit better, takes colors and patterns very well (pinpoint stripes and checks often don't look quite right to me), and tends to shrink less under the duress of commercial laundering (which is how I deal w/ all my dress shirts, and no, ironing them myself is not an option, thanks).
The variations are subtle--I have nothing against pinpoint shirts and enjoy wearing them from time to time, often with tweeds or blazers--but I own more poplin shirts than anything else and the above reasons help to explain why.
PS: FTR, I don't find IME that poplin wrinkles all that much more than pinpoint, and at any rate I think a certain amount of wrinkling is part of the look of a classic cotton dress shirt. Obviously I don't want to look as if I slept in it, but some wrinkling inside the elbow bends or around the waist where the shirt's tucked in simply counts as normal and okay in my book.