cant really see in the video and maybe you are doing all this already but just to confirm....
There are 2 systems that need to be turned off for the sand. With gear in Park you should hold down the stability button for about 4-5 seconds. This switches of both VSC (vehicle stability control) and then TRAC (traction control). 2 separate lights will appear in the speedo: VSC and TRAC. It does them 1 at a time so keep holding the button after the first one lights up until the second comes on too. You need both these systems off for the sand. Or they will grab and slow spinning wheels.
second, and of equal importance, you need to hold for 3-4 seconds the RSCA button (on switch panel next to door) to turn off the roll sensing curtain airbags. Otherwise if you take a side slope the airbags may, anticipating a roll, go off and that will be very expensive indeed. This button does not turn them off completely it seems, it just requires much larger angles to activate. Remember to press and hold to turn them back on when you get back to the tarmac.
Usually I would only use 4L if actually stuck and not moving, and then in 1-2 with little accelerator to turn the wheels very very slowly in order to slowly dig out. Once you are on top of the sand again, stop and select 4H. If you use the manual sequential gearbox, 2-3 mostly, rather than D, you will get much more power down when you need it. tire pressure 12-13 range is good. If you are flooring it in 4L perhaps it is a protection system cutting the power in order to not damage the drive train.