That's why you have the spiral inside the tank, so that it "scoops" the fuel to the bottom.

Not if you spin the rocket end-over-end

But then the rocket isn't consistently pointing the engines in the right direction.

Depends on how fast you need to spin it. If you had a reasonably long period, the lighting of the engines and the cancellation of the angular velocity could occur fast enough that the engines would act like they were consistently pointed in one direction.

I think a bigger issue might be that all the mass is at the engines, so the engines would stay nearly fixed with the other end of the rocket rotating rapidly, and pretty much all the propellant would go to the wrong end of the rocket.

That doesn't sound right. I would expect that the centripetal force will send the fuel to the end of the rocket (or ends, depending on where the pivot point is).