You're absolutely right. There's a fine and subtle distinction to be made here - Freedom of the Press preserve the right of someone to seek to pay for reach for their expression and the right of those with reach to sell it.

As you say, this does not oblige anyone with a press to use it for any given person's benefit.

It just means that the basic concept of paying for expression and to expose other people to it is something baked into the First Amendment. Just not the Speech Clause.