That doesn't look all that soft. I'm guessing your front wheels were in a dip.
Yup, airing down the tires would have helped.

As for the traction, a stiffer suspension would help distributor power better.
And obviously wider tires.

I'd advise slow roll instead of full throttle, but that only works if the other wheels are free to roll.

Looks like fun! Do any doughnuts?