Your view is, to be honest, myopic if not willfully ignorant. Your "freedom of speech" is not specifically dependent on a particular means. However, saying, "You don't need to have access to the Internet in order to express yourself" is a bit like saying, "You don't need to have access to roads, trains, or planes in order to travel." Yes, I suppose that's true, but failing to have access to that travel infrastructure sure does hinder your ability to travel.
It's also worth noting that the First Amendment specifies "Congress shall make no law... abridging the freedom of speech, or of the press". Now a lot of people assume that "freedom of the press" refers to "the press" as in "the fourth estate". However, at the time, referring to "the press" would also have implied simply, "the use of a printing press". This means that the amendment acknowledges that any "freedom of speech" is inhibited by denying access to the technology to disseminate your message.
But let's set all of that aside, because I'm ultimately not talking about a US Constitutional interpretation, arguing that the Federal government is somehow obligated by the First Amendment to provide Internet access to all citizens. I'm not saying that. I'm saying that the "freedom of speech" is not a right granted by the Constitution, but an inalienable right of all people. All people should have the freedom and ability to voice their opinions and views. Regardless of any legal or Constitutional obligations, we have a moral obligation to avoid culturally segregating massive numbers of people, preventing them from speaking their mind in public forums, and preventing them from hearing the viewpoints of others.
And in that sense, and to that end, I think it is a very good idea to provide public Internet access to those who can't afford it. Even if it's not a "free speech" issue in the sense of "The Supreme Court should rule on the Constitutionality of [whatever]", it's a "free speech" issue in the sense of "morally, these people should not be shut out of the conversation."