Yes, the courts believe that independent political expenditures are covered by the first amendment, whether you call them freedom of speech or the press, but that’s not the end of the discussion. It’s well accepted constitutional law that first amendment rights are not absolute.
In already limiting direct political contributions the Supreme Court has already ruled that though they are an expression of freedom of speech (something I don’t agree with) that it is nevertheless constitutional for congress to limit them because of a compelling state interest.
All provisions of the constitution can be reasonably restricted in order to serve a compelling state interest. That includes reasonable restrictions on independent political expenditures even though they are otherwise protected by the First Amendment.
Special interests will only have unacceptably louder voices if we allow them to. They will not have unacceptably louder voices if we don’t.