I am not a physicist but I think here is your mistake. The electron is an excitation of the Electron Field. It's matter aka fermions. The electric field are bosons. Those are two orthogonal things. You can have one without the other (although the fields are coupled).
A boson is not a 'thing' it is a happening. The atom is the first object in physics. That is a shapely thing with location.
Those are two distinct independent things (that can eventually be coupled).
Electricity is mostly a light phenomenon.
With the fields everything happen locally.
If I recall correctly, one of Faraday main discovery was displaying the lines of the magnetic field using metallic powder. Showing that fields were "real" things.
Two electrons don't interact directly with one another. It's more electron interact with photon which then interact with another electron. The one case where two electron interact directly with one another is the Pauli exclusion principle to make sure electron don't find themselves at the same place.
When in doubt follow the energy.
You can store the energy as bumps in light field aka photons (E^2+B^2 (eps0=mu0=c^2=1) ).
You can also store the energy as bumps in the Fermionic field : sums of kinetic energy of electrons.
Finally you can store energy in the coupling between those two fields. But this happen only locally.
At first approximation when dealing with electricity problems what matters is the energy of the electric field, not the kinetic energy of the moving electrons.
In our initial atom at beginning of the vid, the electron has tentacles based on Faraday to account for the tails of the RDF of QM — the indefinite extension of the shell. These will be important in visualizing the atomics of light and gravity in future vids. We ignore them for the circuit because they would obscure the events, but the tentacles remain!
They experience electrostatic repulsion, no electron cloud required.
Local classical Maxwell is enough to explain electricity no need for QM.
I think I get what you are trying to do : "making the Light field implicit". It's a tempting thing to do because when things are coupled we are kind of thinking : is it the electric field which deformed the electron cloud or is it the electron cloud which generated the field.
But this picture is dangerously misleading. It makes you assume strange electron cloud which interact non locally in complicated way (strange arms...). By giving special properties to the electron it doesn't respect the symmetry with respect to charge. Maxwell works just fine for protons. It also completely obscure the facts that we can have external E and B field ; in particular it makes you think that photons need matter to exist which isn't the case.
It is much more clear and general to make light fields and matter fields explicit.
"Light is the way to exchange momentum between charge carriers".
First atoms are neutral, while light affect can only charge particles.
The atom is a composite of a positively charged point-like nucleous, and a negatively charged cloud-like charge density. This complex dance duo, can store energy in between them. Those are the bound states of the electron, but that's not the matter of electricity but chemistry. In electricity, this dance duo can store energy in its surrounding by deforming its electron cloud to become an electrostatic dipole.
Transaction is the wrong picture to have when we are dealing with electricity. The continuous picture is a lot better.
The momentum of an atom is a continuous quantity. At every moment in time it can be exchanged locally in continuous amounts. Both the positive nucleous and electron cloud are taking from the field and giving locally to the field.
Photon is kind of a confusing term because you have to distinguish between the virtual photon which mediates the coulomb interaction in a continuous way, and the real one which can go on its way, or be absorbed/emitted by atoms provided that the electron cloud can deform in such a way to account for energy conservation during the collision.
To see the distinction take the previous example of two protons H+ going towards each other then away. The trajectory to have in mind is they are following a perfect curves trajectories, and not a sequence of straight lines occasionally changing direction when the photon transaction happen. Those typical QM like trajectory you see in cloud chambers need the energy to become bounded in some discrete way. For example electron fly straight, real photon hit and is absorbed and atom change direction to conserve momentum and the electron jump to a higher orbital to conserve the energy (the energy is bounded to the atom for some time), the electron keep flying straight, then it emits a new photon and change direction. These kind of trajectories happen when the energy can only bounded in discrete quantities, but that's a matter about QM, and not electricity.
Finally clarifying what that the light field is carrying in : momentum, and making clear that the light field doesn't carry electric charge.