Burnout 3 is probably the best arcade racer out there. And the game isn't open world.
F-zero and Wipeout would be total shit if they would be open world as well.
Linearity in arcade racers can't be called old-fashioned. Skill-based racing and memorizing the map is what makes an aracde racer fun for me.
If I want to play an open world racing game, I'll boot up GTA V.
One does not always need to exclude the other, though.