Why shouldn't the public be allowed to protest on the sidewalks outside of public official's homes?

Because it's just rude, and someones work life does not exactly correlate with who they are in their professional life.

Why drag their family, the neighborhood into it?

The only reason to do it at a home instead of at the courthouse/legislature/office is harassment.

Because unlike the most influential private individuals, they live in homes with sidewalks - not large gated compounds with their own security details, both designed to make sure the public don't get any chance whatsoever to protest.

Or is that somehow different? If they're politically influential - why, exactly?

Practically, it would mean that you'd have to have a police presence there, wouldn't it? Or are you going to assume that the mob will always play nice, and never decide to storm the house?