There is a tricky thing in the auto take off sequence.
If I remember correctly, you must avoid passing thru other auto modes prior to setting auto take off if you want it to work.
I have my modes set up on a 6 position rotating switch. Like this:
But…I also have a safety switch on my transmitter that locks out the throttle and disables any automatic modes (forcing FBW-A if an auto mode is selected by the rotary switch). The reason for this is very important. In auto modes, even if I have the throttle “off”, the pixhawk can spin the motors. So, a good safety switch setup must disable throttle AND must also disable auto modes! Since my safety switch has 3 positions, I made it:
POS1- Throttle off, manual modes selectable, “Motors off” voiced"
POS2- Throttle enabled, only manual modes enabled, “Manual throttle control enabled” voiced
POS3- Throttle enabled, all modes enabled, “Execute Mission” voiced
If I enable all modes and the throttle and then rotate the mode switch from manual around to Loiter, then RTL, and then Auto, Auto Take Off will fail to trigger on acceleration and I look foolish shaking my plane with nothing happening.
The correct sequence is to leave my safety switch off or at the center position that only enables manual mode throttle. If I then rotate to the auto mode setting and subsequently enable the mode, auto take off will work every time.
4m/s is not a huge push or shake. On my big twin plane, I do a shake and have my plane ramp the motors to full over a 4 second period while I hold it back. At the end, it’s trying to carry me away and it’s time to throw (or just let go).
On smaller pusher planes I just toss the plane and the motor starts after the push has happened and the plane is 5 or 10 feet away.
All my planes have “throttle nudge” enabled. But the throttle is always set to zero during takeoff (since a spinning motor prior to the take off is very hazardous).
There was an exception for using throttle nudge during auto take off. When I did catapult launches, I would advance the throttle a small amount prior to firing the catapult and auto take off. Not enough to move the plane, but enough to have the prop spinning. As soon as I fired the catapult, I would fully advance the throttle. These actions slightly reduced the lag of getting the motor to full speed.