Here is the mechanical analogy to the two capacitor problem that actually fits.
Springs can store energy, Stored energy is proportional to the 'stretch' squared - just like a how capacitors store energy. In this analogy, the spring is our capacitor.
Flywheels can store kinetic energy. They also have very good bearings that cause minimal energy loss. This is our deliberately introduced inductance.
Ropes can transfer the forces (admittedly only when it tension, but we will make sure they stay in tension). Here I am making no statement if these are the wires, or the electric field. These are just a mechanical way to transfer the forces. When the rope is moving it is assumed to have minimal momentum. Consider the momentum of the rope as self-inductance - if the flywheel is very light enough (or things are arranged so no flywheel is needed), this might becomes important in in how the system behaves. But with any sensible flywheel it becomes unimportant.
The flywheel has a clamp on it. When the clamp is on the rope can't move, so no force can be transferred through a clamped flywheel. This is our switch.
The setup
See the lefthand side of the attached image.
We have two springs bolted to the ground, and a very heavy flywheel attached to the roof. We tie a rope to the lefthand spring, wrap it around the flywheel a couple of times so it can't slip, and tie it to the righthand spring, so it is nice and snug, but with minimal stretch on either spring. We check that the springs are both at equal height, and then clamp the flywheel in place.
This is our "zero energy in either capacitor" state.
Charging
With the flywheel clamped, we shorten the righthand rope till we get one unit of stretch in it. We tie the rope off, and trim it off nice and neat.
All of the energy is now in the righthand spring and the apparatus is armed and the experiment can proceed.
The experiment.
We release the clamp. The extra tension of the right hand side causes the flywheel to spin up in the clockwise direction, and the left-hand spring start to stretch.
What I expect to see, in an 'idea' world
When the springs get to equal length, in a perfect world half the energy in the flywheel, so it keeps the rope moving, slowly spinning down, finally coming to a stop when the LH spring now has ALL the energy in it. The cycle reverses, and continues on forever - i.e. the flywheel start turning counterclockwise, and ALL the energy is transferred back to the RH spring.
What I expect to see, in the real world
Energy gets lost due to friction and other effects (like rope slippage and stretch). Regardless of the weight and quality of the flywheel, springs and rope the system will settle down with only half the initial energy stored, shared equally between both springs. The only thing that is different is how long it takes to settle down.
That is it. there is no more to the "Two capacitor problem" than that.