Private financing allows dominance of political process by a minority whose common factor is power to command large quantities of conduct.

Preserving Consent ​​in a Democracy

The Declaration of Independence asserts that

the consent of the governed is required for legitimate government,

as compelling a state interest as there can be.

How does campaign finance relate to consent to governance? 

Private financing of the political process is a problem in a democracy.Two people can have very unequal power at a given moment. Abuse cases hinge on whether a grave imbalance of power between two people impaired one person's free choice. An adhesion contract is not valid because a grave imbalance of power creates doubt that the less-powerful party could not have been coerced.

In a democratic republic just government means representative majority rule with safeguards so that majority rule can't harm any minority.
But an imbalance of power favoring a minority among Americans can make them dangerous to the majority without the converse being true.We don't need laws protecting cars from pedestrians even though there are many more pedestrians than cars. We could see that South African apartheid was unjust even though there were many more black than white South Africans.

Donation requests often break down donations by ranges of amounts, recognizing that at any given moment one person will have more money to donate than another. Although most Americans can donate a dollar fewer can donate a thousand. Premiums increase with range, recognizing that smaller numbers of people can donate the largest amounts. 

Advertising, one form of speech conduct used in politics, has been found to have power to coerce beyond simple influence. And the amount of this conduct someone commands at a given time is almost entirely determined by the amount of money they spend on it. Private financing allows dominance of political process by a minority of people whose common factor is that they have power to command large quantities of conduct. By using the conduct they command to present political speech they can overpower the free consent of the majority of people, who have far less power.
This is a grave imbalance of power that can prevent just self-governance.Because this minority is defined by having more power rather than less, protecting the majority from the minority doesn't endanger the minority.

In order to guarantee just self-government

any imbalance of power in using the political process

must be kept within limits that preserve consent.

Sole public financing of the political process preserves consent,

fulfilling a compelling state interest.