It is essentially the same. Money is needed to effect the speech, especially in the era of professional marketing and electronic media. Of course, you can just stand on top of a soapbox and shout, but how effective would that be? To deliver your speech to the masses, you need money and usually a professional help. I bet the guy jumping from the plane and filming it wasn't working for free in that ad, right? And the plane didn't fly without fuel and pilot, which needed to be paid for, too? This is all necessary components of effective political speech, and by restricting it, you are restricting speech and reducing it to a whisper to be heard only by ones in your immediate vicinity.

>>> If one party spends $100M on their message, and the other just says it loudly at city council meetings

In fact, nothing like this happens. Usually, for any idea worth talking about, there is a lot of money on both sides, because a lot of people in the US have money and would gladly spend it to support the cause they consider right. Much less would be willing to do so if they would face harassment, retaliation and possibly ruin of their life's work for it. I do not think this should be a consequence of you supporting your favorite cause. Do you? If your worst and most corrupt opponents would control the state enforcement machine, would you prefer to be able to work anonymously to dislodge them or not be able to do this without risking your whole life be ruined immediately?

>>> Money is generally a proxy for power.

This is true. But you can not (usually) exercise the power directly against the message - the First Amendment guards against that quite successfully. With privacy destroyed, you can, however, successfully use the power against the messenger or people that allow them to deliver this message - thus effectively killing the message without running afoul of the letter of the First Amendment, but entirely destroying its purpose.