You are confusing some things.
No, you're leaving important bits out, oversimplifying the situation to fit your axiomatic model.
As soon as you close the switch the excess electrons from one plate will move in to wire which is neutrally charged and at the same time on the other plate electrons from the wire will migrate in to the plate with deficit of electrons.
"Will move". No, something makes them move. That something is an electric field, which propagates through the circuit somewhat analogously to a shock wave when the circuit is first connected. Also, some of the original "potential energy" is in the form of an electric field around the charged plate; it is not exactly correct to just lump it all into "potential energy" and call it good enough.
When the circuit has stabilized, the electric field (potential difference along the circuit) has subsided to something small and stable, aside from thermal noise and such. This does not mean it was small and stable and insignificant at the violent beginning.
While electron wave travels through wire
What electron wave? You need to specify that too, and not just give it a name and leave it at that. Giving a thing a name is not the same as describing the thing.
Each individual electron is both a particle and a wave. When an electron is bound to an atom, it is delocalized in the shape described by
spherical harmonics. When an electron is shared by a lattice (as they are in metal conductors), they are delocalized in various ways, and typically spread over or "shared" across multiple lattice atoms.
If we describe electron locations by the center or centroid of their delocalized volume, they really do move very slowly, something like a meter a second or so, often even slower, while the current and changes in the current propagate at over half the speed of light, or over hundred million times faster.
The electrons do not just push each other to move (as described in electrostatic approximation as the Coulomb force); they also interact via emitting and absorbing photons, and coupling to existing electromagnetic fields like the one caused by being matter not cooled to absolute zero. Note that this EM field is NOT just "radiating outwards"; there is always both emission and absorption.
So, what you call "electron wave" is in reality a set of various possible interactions. The majority (i.e., which kind of interaction is the most common or involves the most energy flow) depends on the exact configuration of the system; its geometry.
It is somewhat funny that the most complex phenomena occur when the circuit is first closed, regardless of the current being AC or DC. This case has always wavelike properties, and being non-equilibrium situation, you have all kinds of energy flows all over. Even if we assume a perfect switch, something that changes from 0 to 1 without any intermediate states in between, it still is a step-like pulse with a lot of higher frequency components, and thus definitely a wave. Ramping the current smoothly has a nicer spectrum, but a time-discontinuous signal always contain lots of frequencies.
While the transmission line model does describe the observable voltages and currents at the ends of the line when the properties of the transmission line are known, it does not mean it is a complete picture of the interactions involved. The fact that the model is based on electromagnetic waves, should make it obvious that it is not just about electron kinematics, but electromagnetic field interactions must play a significant role, too.
To explore the complete picture in a way consistent with our best understanding of physics, one needs to delve into quantum electrodynamics, which definitely belongs to the less intuitive section of physics. I definitely have no idea how to even start describing it in laymans terms.