How can an electron have 0 electric potential after exiting a resistor but have current?
If you want to think about the force electrons experience when they are moving in a current carrying conductor, you should not think in terms of voltage, instead electric field is what is forcing the electrons to move (and what is creating the potential difference). In terms of electric field, the Ohm's law can be expressed as
$$j = \sigma E,$$
where, \$j\$ is the current density (per unit area), E is the electric field and \$\sigma\$ is the conductivity. The resistors have a finite conductivity, hence a finite current (through resistor) requires it to have an electric field to force the electrons move in one direction. Due to this field, each electron will experience a force qE and will drift slightly in the direction of force (which is what is called current).
Where does this field come from?
Could come from a battery connected across the terminals of resistor.

Now, assuming the resistor is connected to the battery by an ideal wire having infinite conductivity, from Ohm's law you can see that even if the electric field is zero, you can still have a finite current density. A zero electric field means there is no potential drop across the wire but it still can have finite current.
In reality, the wire does have a finite but very high conductivity so a small electric field (and consequently potential difference) is sufficient to drive current through it. This potential difference can be neglected for practical purposes.