G'Day Alex,
The equivalence of inertial reference frames is what you are missing, I think.
If the sun is truly stationary in space, do you agree that the planets rotate around th sun? Regardless of the speed of "action" of gravity, the sun is in the same place as it was 8 minutes ago or 10 years ago.
Now imaging we are travelling past the solar system perpendicular to the plane of the planets' orbits at a velocity v. We have no effect on the behaviour of the sun-planet system - the planet still rotates around the sun.
The equivalence of inertial (not accelerating) reference frames states this is true, and that it makes no difference if our reference frame was stationary and the sun's was moving.
Relative to our moving reference frame, both the sun and planets have the same velocity perpendicular to the plane of the orbits. They have momentum, and kinetic energy in that direction, and both these must be conserved because we are not applying a force to accelerate any of those bodies.
If the conical model you are thinking about was true, there would be a component of the gravitation force perpendicular to the plane of the planets. This component would cause a change in both the momentum and kinetic energy of the sun planet system, if this was the case in the formative period of the solar system, it can not be sustained without an external force acting selectively on the sun (say the jet out one side I suggested as a thought tool before), so the system would decay to the stable planar orbit we observe today.
I'm not sure I'm explaining myself very well...
Anyway, I hope its giving you something worthwhile and interesting to thin about!
Al.