There are several racing games on older consoles and that were in arcades that I happen to find extremely fun and thrilling. Heck, two of the biggest games from the so-called "golden age" were racing games. These of course being Pole Position and Pole Position II, which I do think are fun, but not exactly all time favorites of mine. But please note that not every game where you drive a car the whole time is a racing game, such as Chase H.Q. for example. But with that out of the way, these are some older racing games I still go back to quite a bit:
Hang-On
OutRun
Super Hang-On
Victory Run
Rad Racer
Roadblasters
Final Lap Twin
Hard Drivin'
Race Drivin'
MotoRoader
Super Off Road
Lotus II:R.E.C.S.
Ayrton Senna's Super Monaco GP II (more specifically the Game Gear/SMS edition)
Daytona USA
Sega Rally Championship
Virtua Racing
Whew, okay, I think that's enough.