If there are other bodies with such a large effect it would be easy to run numerical simulations. The main body, aside from the sun, that has an effect on earth's motion is the moon. As the moon leaves earth, earth's orbit would be affected, but this is not a repeating event. The main motion that has not been satisfactorily explained is the precession of the equinoxes and calling it 'wobble' is probably overlooking an obvious cause.
Wobble does not cover the difference in temperatures. It might effect seismic and therefore geothermal variations, but to dismiss the effect of the planetary gravitational alignments suggests to me that you have not been reading the links I provided.
Right Whale, just one example- the axis of Earth's rotation is indeed precessing and is easily measured by observing Polaris. The circle Polaris makes in the sky is changing.
Back 4000 years ago the pole star was Thuban in Draco, not Polaris.
These orbital wobbles and wiggles are not theory or a model, they are observations.
The theory is that the orbital wiggles can change the Earth's climate- the Milankovich Cycles.