This video shows what happens in a spin.
The turn coordinator shows a left turn, and the slip indicator shows a skid (ball to the far left as well). To get out of it you'd want to push hard on the right rudder pedal.
The yaw damper doesn't allow full rudder deflection.
Also as the instructor shows, pushing the stick left and right does nothing.
The stall protection will push forward, again won't help.
The TBM 930 doesn't have an auto-throttle, so there is no engine control involved either.
The installed Electronic Stability and Protection (ESP) system doesn't handle developed spins—
Leveling at 4,500 feet, I performed a series of steep turns, slow flight, an imminent stall with gear and flaps down, and a power-on stall. As the stall approaches, the G3000’s Underspeed Protection (USP) feature, linked to the angle of attack computer, comes alive as it automatically commands the autopilot servos to lower the nose. ESP, meanwhile, will attempt to return the airplane to level flight if bank or pitch angle limits are exceeded.
The safety features on the TBM would alert and prevent getting into a stall or spin.