Pilots at the controls of the Boeing Co. 737 MAX that crashed in March in Ethiopia initially followed emergency procedures laid out by the plane maker but still failed to recover control of the jet, according to people briefed on the probe’s preliminary findings.

After turning off a flight-control system that was automatically pushing down the plane’s nose shortly after takeoff March 10, these people said, the crew couldn’t get the aircraft to climb and ended up turning it back on and relying on other steps before the final...