Turbopumps powered by rocket fuel require quite a bit of plumbing and design work; you're essentially building a second rocket motor to drive the bigger one. By using an electric turbopump, you just need a place to put the motor and batteries, which is a lot more simple in comparison.
Yes, the turbopump is run off its own rocket engine --- which drove a 40MW turbine, which ran a pump that could push three tonnes of rocket fuel a second.
Admittedly, the Saturn V was a little bit bigger than this rocket's going to be, and had special needs, but it goes to show that rocket fuel powered turbopumps are scary.
(Incidentally, SpaceX use a rocket fuel powered turbopump for the Falcon 9. I believe the scary-looking plume of flame that comes out sideways is the exhaust. It's a mere 2MW. Per engine.)
