Spinning rockets are extremely hard to control due to precession/conservation of angular momentum. Any change in attitude would introduce a wobble. Unless you make your tanks spherical and rotating independently of the rocket motors on a gimbal you will have trouble with changing orientation and compensating for minor offsets of thrust.

Ah, you're right. You'd need to spin until just the moment when you start the engines, and then stop pretty much instantaneously, which in practice would cause all kinds of instabilities and problems...

Spinning the rocket sends the liquid to the sides, not the bottom where the pumps are.

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).

And how do you spin up the rocket?

Using a reaction wheel.

https://en.wikipedia.org/wiki/Reaction_wheel

Basically, an heavy enclosed wheel that you use a motor to spin. Because of the conservation of angular momentum, the rocket will spin in the opposite direction (more slowly, because it's (probably) heavier).

Not going to work at that scale.