My parents have used a laundry chute made of rectangular sheet metal
ducting for years, between two studs in the wall (therefore only 3 or 4
inches deep), and it's worked fine. No problem with clothes snagging.
If you'd rather use wood, I don't know why 1x3's wouldn't work - it
shouldn't take a lot of abuse, right? Also, if I understand your plans
correctly, it looks like the frame extends all the way to the floor of
the basement? My parent's chute just extends a few feet through the
basement ceiling, and the clothes obviously fall straight down from
there; no need for a longer chute.
Good luck,
Andy