Race car tires are meant to go a few miles before you throw them away, and are carefully inspected before each session. Passenger car tires are meant to go tens of thousands of miles before you throw them away, and many people almost forget their car has tires until one goes flat, hopefully not at-speed. You can look at a race car tire before you drive on it, every time you drive on it (i.e, once) and verify that there isn’t any structural damage. While you can technically do that to a passenger car tire, no one does including I’d wager the gearheads here.
If you get into a scrape with a race car tire that damages the cord, you’ve probably also damaged the car to the point that it’s coming off the track, often on a tow strap. Crashing at 150mph tends to do that. If you get into a scrape with a passenger car tire that damages the cord because the rubber is gone, you might have been doing all of 5mph when you scraped that curb, and you won’t think anything of it until you’re doing 70 and the tire blows.
Inspectors do not tend to fail cars because they aren’t all pretty and ready for Pebble Beach - they fail them because of safety and/or environmental defects that need to be corrected sooner than later.