"die" isn't really enough to translate "that". "Wir lesen die Zeitung" = "We read the newspaper".
Colloquially, you can just use the definite article to point out "this/that object" with grammar intact: "Welche Zeitung liest du?" "Die Zeitung." ("This newspaper.") - but even then, you'd more often add "hier / da" ("here / there") for some emphasis: "Die Zeitung hier / da." And you can leave out "Zeitung" to make it shorter: "Welche Zeitung liest du?" "Die (hier / da)." ("This/that one.")
Officially, though, you definitely need to translate "this / that" with "diese(r/s)". (Also: "Welche Zeitung liest du?" "Diese (hier / da).")
And officially, you'd have to translate "that" with "jene(r/s)", but it's hardly ever used (even in writing), and I'm not sure I've even come across it on Duo.
I'd say that translating "that" with jener would be wrong, precisely because it's not used in German.
It may be useful for Germans who are learning English to differentiate "this" and "that", but "that" is best translated with das (da), in my opinion.
On the other hand, translating "that" as diese seems wrong to me.
I think we agree that with "jene(r/s)" gone from today's German, it leaves an empty space where the English "that" would be.
In real life, I'd say "this" and "that" aren't as strictly separated as you're taught at school, with "this" = here, "that" = there. "This" is still "here", but how far away does an object have to be in order to warrant the word "that"?
I think it's a lot about (more colloquially) spoken language vs. (especially written) more formal language.
In spoken language, I do use "dieser", but more frequently "der da", no matter how far away the object is. Theoretically the distance can be varied by using "der hier" (close by), "der da" (quite neutral, but more like "that"), and "der dort" (far away), but personally, in real life, I don't bother with them.
In written language: "After crossing the Misty Mountains, they see a peculiar mountain far off at the horizon. That mountain is Mount Doom, and they need to go to that mountain to destroy the ring in that mountain." You wouldn't normally use "der Berg da/dort" unless you're in a position to literally point it out for somebody to see. You wouldn't just use "der Berg", because it lacks emphasis (because in writing you don't see a difference to just "the mountain"). You would use "dieser Berg", even if it's far away. Basically because you can't say "jener Berg" without sounding like a grammar book (or very very old), as you rightly said.
This conversation reminds me of a story my Dad tells. His family is German so he speaks German fluently. He's also an international business lawyer. One time he had to give a speech in German so he sent it to the daughter of family friends. He says to her that she wants it to read like he was talking to kids her age. When he got it back she says 'You sound like a German Austin Powers!'