Having trouble intuitively understanding why complete circuits are necessary.

Remember; for every action there is an equal and opposite reaction.

As others have mentioned the analogies can sometimes be more confusing.

Consider the water analogy. Here the common force is gravity. If you imagined the same experiment setup in space without gravity, the result would be much different.

For lightning it is similar, the circuit is completed by a large (physically) capacitor from the air at the cloud level to ground.

If this were the case, then why can't you run a wire from the negative end of one battery to the positive end of a second battery and expect current to flow from the electron-plentiful negative terminal of battery 1 to the electron-deficient positive terminal of battery 2?

Here the lack of "electron flow" is due to there being no difference in potential. Battery 1 negative is at the same potential as battery 2 positive (by the fact that they are connected together).

I'm not sure I helped but I hope you can "see" the completed circuits in the analogies.