It ain’t easy—or cheap—to be a car guy. At bare minimum, you need a few grand in disposable income, a few square feet of accessible free space to park, a few tools, and a little mechanical know-how.

Both of GM’s performance-oriented divisions targeted 1963 road racing to bolster the images of new, underwhelming economy cars. In February, Paul Goldsmith led every lap of Daytona’s 250-mile American ...