One can write a sentence with a partial quotation:
Direct speech: We live in a madhouse! We have to move.
She says they "live in a madhouse" most of the time.
This is unobtrusive: the quotation is direct speech, because the exact words are repeated, made to fit into the syntax of the sentence, which was easy enough. It is a predicate, so it fits together with the subject they and the adverbial phrase most of the time to form a complete clause.
Direct: You know that they like it here.
She says that they "like it here".
Similar: no problem.
?She says that "they like it here".
This would also be direct speech, because the exact words are repeated. It is made to fit within the syntax of the sentence. This style is perhaps legitimate, but it is, in my opinion, inadvisable. I would prefer not to repeat they in the quotation, but rather keep that word as part of the embedding sentence, and only make "like it here" the quotation, as in the first option.
At any rate, with indirect speech, it is essential that the quoted part would fit into the sentence even if you removed the quotation marks (you'd only lose the knowledge that it was a quotation: otherwise, the meaning should not change).
?In 2000, Robert Scheer created the website New Age Journal, which states that "We are not affiliated with any magazines printed on paper."
I see two problems here, if we assumed the quotation marks were erroneous and we read it as indirect speech. First, there is a capital letter, which makes it not fit the surrounding sentence. Secondly, we would not make sense in the main sentence: in the main text, there is no we speaking, just Wikipaedia.
Conclusion: you are correct that this quotation can only be read as direct speech, and I agree that there should be no that before a normal quotation of direct speech. In addition, I would probably use a comma immediately before the direction quotation, or possibly a colon?