Painius wrote:

"A greater force of gravity on its sides" indeed does mean what you
are calling the radial components. They stretch from the axis of the
tunnel out in all 3 dimensional directions toward the Earth's surface.
Disregarding the reality check of other frictional effects and just
focusing upon these radial components of gravitational force, i do
not believe that they will cancel out entirely.

If Earth were a sphere whose surface were smooth and whose
innards were uniformly and consistently dense, then yes, the radial
components would cancel out. Since this is not the case, since the
Earth's surface is not perfectly smooth and since its inner volume
varies in density, then these radial components would not cancel
out and would have an effect on anything that oscillates within the
tunnel. The effect would then be a sort of gravitational "friction"
that would dampen the oscillations. And this (i realize that i'm
being loose with my terminology here) gravitational friction would
also affect every cubic cm of material within Earth's surface,
wouldn't it?

I agree with you up to where you say the imbalanced radial forces
would cause damping. I think they'd just perturb the 'orbit' a
little. If the walls of the tunnel are frictionless and perfectly
elastic, even if the car goes 'off course' far enough to collide with
them no energy will be lost. Of course if those conditions aren't met
there will be a damping effect.

Putting it another way, for conservation of mechanical energy to be
violated there must be "dissipative" forces at work, and AFAIK
gravity is not one of them.

--Odysseus