Does the hero have to be on a bike? Snatcher features a section where Random Hajile has to catch up with Gillian's out of control Turbocycle, via his motorcycle, and Policenauts has a shoot-out with a motorcyclist from Jonathan and Ed's speeding car.

No, I don’t think so. Should probably add Resident Evil 5 too for the same reason.

midnight fight express and crash bandicoot warped come to mind

My Friend Pedro?

Sure!

Astral Chain has a couple of motorcycle combat stages, and Mario Odyssey had you racing towards the screen on a vespa while being chased by a T-Rex