We stayed in national forest campgrounds all the time and never found a 2-night requirement. Of course, being that they're mostly in the boonies we never stayed for only one night. We were there for a purpose - to explore the area, not a quick drive-thru. Also, we didn't travel on Fri or Sat so we didn't make reservations... always got a site mid-week.
I guess if you need to travel on a Friday on your way to somewhere and want only one night the best thing to do is to stay at a parking lot enroute and leave early the next morning. We have also found that there are some very nice small-town city or county parks but on a Friday you'd have to get there early afternoon if you didn't have reservations.