Okay, so. Here's my hypothesis about the conspiracy theories—why people want to believe that the Supernatural writers *wanted* to write Dean and Castiel as explicitly queer and in love, but were prevented by the network. (Thread)
Fans of this show have clearly poured a lot of energy and emotion, over many years, into interpreting it and guessing at where the story and characters would go. As is generally the case with long-running shows like this one.
For a significant chunk of the fandom, a lot of that effort has been directed towards interpreting Dean and Castiel as a) both queer characters and b) specifically in love with each other, and in finding things in the text to support that interpretation.
For the sake of brevity, I'm going to refer to "things fans point to in the show as evidence that Dean and Castiel are queer/ in love" as "clues". The term also fits with the "looking for hidden messages/ figuring out what's going to happen" that seems to be going on here.
So, in this scenario, the fan believes that the writers of the show:

a) interpret Dean and Cas as queer characters
b) interpret them as romantically attracted to each other
c) intend to, eventually, make both of these things explicitly obvious on the show ("canon")
The fan, therefore, looks for clues in the show that support these beliefs. A lot of interpretive writing about the show ("meta") on social media involves finding these clues, in anything from dialogue to set decoration and wardrobe, in words and in symbolism.
This is a lot of work, for one thing. It takes time and energy. It's also likely to be very emotional for the fan—there's a good chance they identify as queer, and therefore something of their personal identity is bound to these fictional characters they interpret as queer.
So you get to the end of the show, and there's no explicit textual acknowledgement of these beliefs. It's just as vague and subjective as everything that came before. There's no definitive, objective proof that these are queer characters in love; it's left open to interpretation.
This is where I think the appeal of a conspiracy theory comes into play. The fan has put so much energy and emotional investment into the idea that queer/in love Dean and Castiel would be canon, that all the clues were pointing towards that being the endgame.
But it doesn't happen. So they're forced to think: Was I wrong? All this time I thought I was picking up on clues the writers were seeding in to hint at this becoming canon. Was I just imagining it?

That's upsetting. Maybe they feel stupid for thinking all that.
Enter the conspiracy theory: The writers WERE leaving all those clues. They DID intend to make this canon. But something stopped them (the network) and they never got to pay it off. Maybe they knew they'd be stopped, but they left in the clues so you'd know what they'd intended.
This is appealing, because it preserves the validity of all the clue-hunting. The fan doesn't have to feel stupid for seeing all these clues, because they WERE real. It also preserves the idea that the canon queer characters/ship they wanted exists; it was just suppressed.
So that's my best working hypothesis, at the moment, for why people are attracted to this conspiracy theory.
So, what to do about it?

It's fair to ask, "Why do anything about it? Let people believe what they want. Let them cope however they need to cope."

The issue is, conspiracy theorizing isn't coping, and it can be harmful. Even if it's just about a TV show.
Sure, it's not harmful in the way political conspiracy theories are; those are more likely to drive people to take harmful, real-world actions against others.

But conspiracy thinking can be personally harmful to the person doing it.
One issue with conspiracy thinking is that it doesn't seem to give people the things they need that drive them towards conspiracy theories in the first place. (If they're motivated by feeling powerless, for example, conspiracy thinking seems to make that WORSE.)
Another issue is that it can affect behavior in counterproductive ways—for example, if a person believes malicious people/groups have disproportionate power over what happens, they may be less likely to take control in their own lives. (e.g., "Why vote if the vote is rigged?")
Therefore it seems like you *can* counter conspiracy theories with refuting evidence, if you also take care not to trigger that defensive response by making them feel like you're attacking them personally on this aspect of their identity.
Again, I think the likeliest, simplest explanation for this situation is that the writers never had any intention of making this ship canon. However, I also think the "clues" people were seeing were still really there, and I think that may suggest a way to combat these theories.
Because honestly? I think the clues, at least some of them, were intentional. I think the writers of this show knew damn well people were interpreting the characters this way, and they encouraged it because it kept people interested.
TV writers in general engage in a lot of ship-teasing, and that's generally fair play. But if you don't intend to pay it off ever, it's kind of a dick move. Especially when queer ships, which are underrepresented in TV, are involved.
And I think that's what happened with Supernatural. I think it WAS "queerbaiting", in the sense it's used re: media these days to mean "hinting that you're going to make a character canonically queer/ a queer ship canonical, when you have no intention of actually doing so."
And that's shitty and infuriating, but I think that's the reality that Supernatural fans have to face, if they believe they really were seeing clues that pointed towards Dean/Castiel that the writers really meant to put there, but didn't pay off canonically.
So that's my suggestion. Face that, instead of promoting much less likely theories that the CW—a network with several other shows that have canonically queer main characters and queer ships—suppressed canonically queer characters/ships on THIS show.
Deal with your anger about THAT. Talk through THAT. Don't redirect it to conspiracy theories, because among other things, it poisons your thinking in a general sense. And god, the last thing anyone needs these days is to be MORE susceptible to irrational thinking.
Don't promote conspiracy theories. Talk about what is much more likely to be the case—that a show that jerked you around about queer rep for 15 years kept doing so right up till the end. Be mad about it, if you are. Be upset about it, if you are.
And then go out and make better art. Whether it's based on this show or something else, or something original. Be inspired by your anger and turn it into something better. Be the canon queer rep you want to see in the world.
And consume other art. Watch shows and movies and read books and play video games with canonical queer rep, that don't jerk you around like Supernatural did. There is SO MUCH fiction out there with canonical queer characters.
Make better shows. Watch better shows.

Don't retreat into conspiracy theories to avoid dealing with reality—make reality suck less.
Appendix:
• I am not a mental health professional and this is not medical advice
• I did research and reading on conspiracy thinking in undergrad; I am hoping to do more in postgrad
• I was in the Supernatural fandom for ~5 years so I am not wholly unbiased here
There is lots of pretty badass research on conspiracy thinking out there; check out the works of the researchers cited in the articles I linked. If you want some papers to read, I have a bunch I can send you
If you want to make TV that doesn't suck:
• Learn TV writing. This stuff comes highly recommended: https://linktr.ee/OKBJGM 
• When you are ready, look into writing fellowships like Disney/ABC, they've launched lots of careers
• Be gay do writing
You can follow @madseance.
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