That Connew F1 story was neat! Too bad it's not possible today.
The other quasi-amateur privateer that comes to mind as a self-builder is Alain de Cadenet.
To quote Wikipedia:
He made his reputation building and driving his own sports prototypes, taking on works teams and occasionally beating them. In 1976 he finished 3rd overall at the Le Mans.[2] In 1980, with co-driver Desiré Wilson, he won two rounds of the World Sportscar Championship —the Monza 1000 kilometers and Silverstone 6 hour events. This was a major achievement in an era of increasing professionalism, when it was very difficult for privateers to defeat larger, better-funded teams that had factory support.