Should it? Should free speech be extended beyond the bounds of government and if so doesn’t that contradict its original meaning by making the government restrict who should and shouldn’t provide uncensored free speech?
Free speech never began with governments. The 1st is a prohibition on government interfering with the freedom of speech. That's the problem with this whole debate, most people have everything backwards. Free speech is a concept, an ideal that underpins the entire concept of a healthy, free, fair and open society.
> What about slander and libel laws or laws about claims made in advertisements?
Yep — here we're getting into Prior Restraint. It's both against the law to restrain free speech such as lies, but as you rightly point out, this speech can have other legal consequences.