Motorsport isn’t lacking when it comes to cameras. Formula E — the racing series where all 11 teams and 22 drivers throw fully electric cars around iconic tracks and street circuits at speeds of 200 miles per hour — outfits each racecar with nine onboard cameras, providing different views from the nose to the tail of the car. But for the new Prime Video series “Formula E: Driver,” the production company Astronaut films wanted to push the limit of how an audience can experience what it’s like behind the wheel.

The sports docuseries follows four drivers in particular — António Félix da Costa, Mitch Evans, Jake Dennis, and Dan Ticktum — across their battles in the ‘23-’24 campaign. And while the overall approach to being trackside during races without disrupting the teams is similar to Netflix cousin “Drive to Survive,” “Formula E: Driver” has a couple of things that make it distinct. One is that...