You're making an incorrect assumption: electrons don't gain kinetic energy as they travel through a section of wire - not from the negative pole to $A$, and not from $A$ to $B$. As they travel through any section of wire, they gain some kinetic energy due to the electric field, but then they smash with an atom and lose it all, then they gain some again until they smash with another atom and lose it all again. And this repeats over and over Overall, however, their average kinetic energy (and thus average velocities) stay constant through any section of wire. You can prove this to yourself by seeing that the current is constant throughout a single series section of wire.
First of all, look at your first two lines (with a few edits just to make them more grammatically pleasing, at least compared to the way in which they were first written):
The electric field through a closed series circuit is constant. This
means that the force on each electron moving through the circuit is constant. According to
Newtonian mechanics, the total work done to the electrons as they move from position A to position B is the difference in the electrons' kinetic energy at position $B$ and position $A$.
I'm gonna stop you there. Think about this: if what you're saying is true, and there indeed WAS a change in the kinetic energy of the electrons between "position $A$ and position $B$" in the circuit, that would mean electrons would be moving faster at position $B$ than at position $A$ (assuming they're moving from $A \rightarrow B$), since they gained kinetic energy.
Is that what truly happens?
The current throughout a circuit is constant, so the electrons CAN'T have more kinetic energy at position $B$ than at $A$. A change in the velocity of the electrons would imply a change in the current through the circuit. Since the current is constant throughout a series circuit, we know the electrons can't have sped up - they can't have gained Kinetic Energy.
The work done on the electrons by the electric field doesn't increase the kinetic energy of the electrons - it simply counters the work done on the electrons by the atoms within resistors, atoms that the electrons are constantly smashing into. The work done by the resistors on the electrons cancels out the work done on the electrons by the battery, causing the electrons to maintain a constant velocity through the circuit.