Purpose of K+ channels in action potential
Great question! However, your question is based on some misconceptions about what polarization means and how ion movement is involved, as well as the difference between equilibrium and the time it takes to get there. That's okay - it's a mistake that many many people learning about neurophysiology make, including instructors.

Na+/K+ concentrations actually change very very little during an action potential. The Na+/K+  pump establishes the concentration gradient, not the resting membrane potential. The resting membrane potential is caused by the concentration gradient plus the relative permeability of different ions via the Goldman equation: specifically, that permeability to K+ is much much higher than Na+.

When you open voltage-gated Na+ channels during an action potential, the relative permeability to sodium increases dramatically, so the new "equilibrium potential" is closer to the reversal potential for sodium (again, use the Goldman equation). There will be a net flow of ions (mostly Na+ coming in) until that new equilibrium potential is reached.

However, voltage-gated Na+ channels inactivate quickly, so that equilibrium isn't really reached. Instead, as the Na+ channels close, the new equilibrium potential is back near the original resting potential (for a third time, use the Goldman equation), because the primary permeability is to K+ through leak channels, and therefore the net current flow direction changes.

Okay, so what's the need for the voltage-gated potassium channels in all this? Well, the rising phase (sodium phase) of the action potential is fast because there is a really high conductance to Na+ so lots of Na+ can flow quickly in. In comparison, there is a lot less conductance of K+ through the leak channels. If all you have are leak channels, it takes a long time to return to rest. Equilibrium potential is just that: the equilibrium. It takes time to get to equilibrium. Voltage-gated K+ channels speed up that return to rest.

What if you had no K+  conductance, including no K+ leak channels? Well, then it doesn't matter at all what the relative concentration of K+ is inside and outside the cell: check the Goldman equation! You can't really have no conductance to anything at all, but if there is no dominant ion then the membrane potential will be something around zero. In that condition, the charge imbalance of the Na+/K+ pump process can contribute a couple mV, but nothing more than that.

All that said, to address your specific questions:


  I understand that they serve to repolarize the neuron after the Na+ influx. What I don't understand is why this is important.


It's important for speed and for the minimum refractory period between action potentials. Nervous system activity would slow way way down if action potentials could only occur every 100 ms.


  Instead, we have to wait until the Na+/K+ pump can restore the proper intra-/extra- cellular balance between them. And then the neuron can fire again.
  
  But don't we have to wait until the Na+ concentration is restored before we can fire again anyway?


Na+/K+ pump is only needed to maintain ion balance over a very long term. On the time course of a single action potential, the concentrations of Na+ and K+ do not change appreciably. That's not what repolarization is.


  So what do we really gain by having a way to restore the charge differential?


Speed!

You can also see my answers on some related questions:

Why is it possible to calculate the equilibrium potential of an ion using the Nernst equation from empirical measurements in the cell at rest?

Why does K+ moves out of the cell?