Identifying That N64 Futuristic Shooter with Vehicles & C-Button Controls: It's Blast Corps

You're almost certainly thinking of Blast Corps (1997, Rare / Nintendo 64).

• Futuristic Action/Shooter: You pilot demolition vehicles to clear a path for a runaway nuclear carrier.
• Diverse Vehicle Roster: Bulldozer (Ramdozer), excavator (Backlash), crane (Skyfall), muscle car (J-Bomb), motorcycle (Ballista), dump truck (Sideswipe), and more.
• Heavy C-Button Usage: The three yellow C-buttons controlled gear shifts (high/low), special attacks (e.g., J-Bomb's jump), and camera/aim—core to the control scheme.

If this matches your memory—destroying buildings with a backflipping dump truck to save a nuke carrier—it's definitely Blast Corps.