This is where you go wrong, because that's not technically true because there is textual evidence that others believe Kari dead: there was no big conspiracy in the Two Rivers to keep Rand in the dark about his early years or his mother. Tam hid Rand's real origins from everyone in EF but Kari, because he believed Rand was "an Aiel savage". The Aiel weren't exactly well seen after the Aiel War, Rand would have faced all sort of prejudices, and on the way back to EF he may even have been in danger if it was known he was the child of an Aiel woman (sic). Tam and Kari wouldn't have taken that chance anyway.

We know Tam suspected nothing about the DR. It's the fact he was Aiel Tam wanted to keep even from Rand himself. That's why he hid the fact he was adopted (there's fairly obviously no prejudice against adoption in the TR/Andoran culture). Whether it's because he disliked lying or feared he'd make a slip, and perhaps because as a veteran he didn't like to talk of those years, Tam appears to have discouraged all questions about his time out of the TR, period. Rand was close to his father, but knew better than to bring that subject up (as he pointed himself in his surprise that Tam brought it up himself about the sword).

Nynaeve was hostile when aske about Rand's origins not because of any big secret around the TR concerning Rand that she knew about, but because she distrusted Moiraine, and wasn't forthcoming with Rand because she didn't want him to go and start believing Moiraine's stuff. It wasn't a secret that Rand was brought to the TR by Kari and Tam, it's just something Rand never asked about.

So "people" don't believe Kari is dead only because Rand thinks so, but from all sort of textual evidence. Perrin, Egwene and Mat were very young, but Nynaeve remembers Kari quite well (and forcibly, her death), even Tam's return to EF with her and baby Rand. Tam's deeply marked by Kari's death, and it's a well-known fact around EF. Rand heard all sorts of whispers from goodwives that Tam has been widowed too long and should remarry. Right there it's textual evidence that the people of EF know Kari is dead - just think of how Nynaeve reacted to the notion of polygamy later on..... When Kari got sick (or after she died) the wisdom would have been called, funeral rites would have been performed. The whole village knows.

"Kari being alive" isn't a "POV trap", you're talking about this big conspiracy to make her vanish and trick the whole village to think she has died. Unless this conspiracy involved Tam, Mistress Barran and others (and what on earth would have been their motive for this, really?), who exactly got buried in Kari's place? Cultures like the TR are still very "do-it-yourself" with matters like death. You can be sure many saw the body. The Wisdom would, and goodwives who came to dress Kari up for the burial, and the people who put her in the ground or in a coffin etc. If her death wasn't sudden but more like a wasting sickness, she probably died not at the farm but in the village, at the Wisdom's house maybe, or at the sickhouse if there were fears that she had could be contagious. And how do we even know that Rand didn't see his mother's disease and death? Nothing in the series states that he didn't. Why do you assume he's been told (and lied to) about it?

So you've got Rand, and Nynaeve, and the goodwives of the village, including Egwene's mother who is mentionned specifically about her less pushy attitude toward Tam's widowhood. And Tam's attitude clearly show he believes his wife dead. All the village know Kari is dead, it's not just Rand.

What made you believe such a conspiracy about Kari existed anyway? Can't be the Shadow, the Shadow learned about the three boys long after Kari died, and about Rand being the one only in EOTW itself. Can't be about the fact Rand is the DR, Tam suspected nothing.