> It's important for people to remember they don't marry other people.

Of course you do. As you even point out yourself, you marry the government, which is people.

> The first time I came across this it made me wonder how much people are aware of this central, third party in between the two people in their marriage.

Considering that the traditional marriage process involves two people standing in front of representatives of the government and declaring promises to them it seems like it'd be a bit hard to miss? I suppose if others also don't realize that government is people then they might not realize what the people in the audience are doing? But, then again, given that the two people prepared to make those promises are usually responsible for inviting that audience to the wedding you'd think they would have questions about why they are there otherwise?