It sounds like you want a racing simulator, and not just some arcade racer. Games like Grid and NFS:Shift are weird because they offer more realism then your average arcade racer (NFS: Most Wanted for example), but they still don't offer you the realism that a true simulator would offer.
I can reccommend rFactor as a great sim to get if you are looking for a customizable platform where you can race built in cars, or race in all sorts of mods to your hearts content. rFactor does have single player, but it is more multilayer focused. In my opinion, it is much more fun to race against real people then against dumb computer controlled cars.
Other good racing simulators are the Simbin games. These are GTR2, GTR Evolution, Race On, Race 07, etc. These games can all be found on steam. They probably have a little better single player then rfactor. To me though, it seems like there are more mods and more variety of racing types available in rfactor then in the simbin games. Also, rfactor feels more realistic to me then the simbin games. However, they are all still fun.