The definition of capitalism is the private ownership of production and its use to generate profits.

I think a coerced assumption you may have of capitalism is that corruption is an unintended side effect, but it actually follows from its principles.

How is a society to maintain unmarred democratic institutions when its elements are fundamentally unequal? Put more clearly: How can people have the same amount of political power when one class (capitalist) OWNS the production of what the other class (workers) need?

The mythology of capitalist society paints both them as equals and the state as neutral. This is a tactic to preserve the appearance of a democratic backbone. They afford this mythology because capitalists own the air waves and they have, and can have, the most influence in the state. In fact, due to this fundamental inequality capitalists are, for all practical purposes, capitalists are the state.

Capitalist societies put political power up for auction; Corruption has its highest manifestation within capitalist societies.

Now to your point. Greed will never "magically melt away". Greed can only be controlled through democratic control of what permits greed in the first place.

Communism/socialism isn't about magically doing or undoing anything, it's the science of creating firm and unalienable working class power. It must start with democratic control of production and local peoples councils. Greed will not magically melt away, greed must be constantly cut out by everyone by everyone HAVING the political power to cut it out. This means peoples councils will be convened at the neighborhood level, peoples courts will be manned, not by professional judges, but by rotating locally elected citizens. Council delegates will be bound by law to only, and exclusively, be messengers at higher level councils, etc. This is just a small picture of what democracy is. It is not me to say specifically how, of course, but communism does not involve blindly and powerlessly trusting political candidates, like capitalist society requires.

There is a reason communism is demonized by the people who control our society.