Was the automobile and plane invented at close to the same time?

Carts and kites were invented a long time ago. The significant invention was the internal combustion engine. What we call the automobile was first produced in 1885 by Karl Benz and Gotlieb Daimler. Powered heavier-than-air flight was achieved by the Wright brothers, but lots of people flew gliders before that.

The Chinese did not invent an airplane because two Wongs don’t make a Wright.

Your basic premise is flawed. People don’t die in auto accidents or most plane accidents because of mechanical failure. They die of operator or pilot error. (design failure such as the SuperMAX 737 problem is so rare, that is why there is extraordinary focus on it.)

The first production car dates to Benz’s auto, 1885.

The first heavier than air flight was December 17, 1903, eighteen years later.

Air travel is by any measure the safest way to travel.

Cars are vastly improved over the way they were 50 years ago. Auto deaths have declined as cars become safer.

Here is a chart of traffic fatalities. Deaths related to miles have dropped dramatically and without apparent relation to any particular improvement in roads or equipment. Further search turns up data indicating that about half of fatalities involve a single vehicle and either a tire failure or a collision with a fixed object.
https://en.wikipedia.org/wiki/Motor_vehicle_fatality_rate_in_U.S._by_year

Invention of the internal combustion engine.