Driving over the speed limit is speeding, therefore actionable. Driving under the speed limit indicates impairment, therefore actionable. Driving exactly at the speed limit indicates a wise-ass and we can't have that.
I just finished taking a defensive driving course to get a speeding ticket washed.
The course material mentioned a couple of times that, quote, "driving at an unsafe speed" is illegal. And what's unsafe takes into account the prevailing conditions (so you can be ticketed when going the speed limit if, say, it's raining), and also the fact that deltas in speed are more dangerous than absolute speed (so if the speed limit is 55, everybody else is going 60, and you're going 40, you may well get a ticket 'cause you're creating a danger.
Almost sounds like the police can pull over and search whoever they like
Yep, that's true within 100 miles of the border - an area that accounts for the vast majority of US residents. I've been stopped several times near the Mexico border for no real reason; one office asks me questions for a minute or two, just long enough for their drug-sniffing dog to make the full round of my car and come up empty.