The key distinction is between thermodynamics and kinetics. "Feasible/possible" is a bit vague and doesn't distinguish between thermodynamic and kinetic feasibility.
Thermodynamic feasibility is about the energy difference between the reactants and products. (Strictly that should be "free energy difference", but you might not have come across free energy at high school level. If you have, take what I say about energy as referring to free energy. If not, don't worry about it, but realise that it's not strictly correct applied to energy, but free energy will be explained to you in due course, if you continue with chemistry.)
If the energy of the products is lower than that of the reactants, the reaction is said to be thermodynamically favourable (i.e. the equilibrium constant is >1). Kinetic feasibility is about the rate of reaction, which depends on the size of the energy barrier that has to be got over for reactants to transform to products. This is called the activation energy, and the important point is that there is no necessary correlation between the (thermodynamic) energy of reaction and the activation energy.
This is illustrated in the attached diagram. ΔH is the enthalpy (energy) change for the reaction, i.e. the difference between products and reactants, which determines thermodynamic stability (strictly should be ΔG, free energy). Eaf is the activation energy for the (forward) reaction, which determines the kinetics. A reaction with a very high activation energy may be kinetically extremely slow, even if the reaction is highly thermodynamically favourable.
Electrode potentials are about equilibrium situations, therefore about thermodynamics, not kinetics. The point about Li and Na is that the reduction of Li is more thermodynamically favourable than that of Na, but the reaction of Na is kinetically faster. A simple (but probably incomplete) explanation is that the ionisation energy of Na is lower than for Li, for the reasons you suggest, but the solvation energy of the Li+ ion is greater than that of Na+, as pointed out in the link posted by thetada.