Is Prodigy still canon?

It depends on whether or not you think the act of distributing is part of the commonly understood definition of canon.

The definition most reasonable Trek fans operate on is "the shows and movies made by the rights holders," although most aren't aware they've internalized that second part because nobody wants to admit they spend any amount of time caring about "the rights." (Ask them if they think Continues or New Voyages is canon and you'll cut to the heart of that matter real quick.) That latent "rights holders" qualification isn't there out of any particular deference to Paramount, it just gives us a convenient and durable boundary that a huge, varied, and global fanbase can largely agree on. Paramount decides what Star Trek to produce, but that decision results in canon Star Trek because it's a simple enough boundary for Trekkies to collectively accept without much friction.

Point being, Paramount is part of the definition but Paramount is not the source of authority for the definition. The fans are. This is an important distinction if you want to investigate whether or not this ugly Prodigy business has altered or clarified the definition of canon.

This act of cancelling and archiving is uncharted territory, yes. Assuming you agree with my earlier definition, it comes down to whether or not you think there are more qualifiers hiding after the word "made." Something like "made, released, and currently being distributed," which is an interesting set of qualifiers to add because it would decanonize pre-remaster TOS.

Personally, I think that's too much control over the definition to hand to Paramount. We care about "made" because it's the most minimal way to establish this "rights holders" boundary. I think we could have a reasonable debate over whether or not "released" is already part of the intuitively understood definition (i.e. "are deleted scenes canon?") but I would wager most Trekkies will agree that "currently being distributed" is not part of the intuitively understood definition.

So, yes. Prodigy is still canon.

I believe this post is operating under a misunderstanding. You can still watch Prodigy. It’s available for sale on iTunes right now. Until DVDs became widespread, it was often difficult to watch previously aired shows, especially in a complete way, but TOS and TNG never became non-canon and Prodigy is infinitely easier to watch than those days.

It's canon until it's not (ie, explicitly contradicted by some other Trek installment). And even then, canon in fiction is rather a silly concept anyway, and is largely more up to the collective opinion of the fan base than whatever big corporations owns the rights to it.

The problem with the question is that it’s based on the assumption that TAS was not considered canon because it was unavailable, which is in error. TAS was decanonised in fandom’s minds only because Roddenberry decided he didn’t want to consider it canon, not because nobody was watching it.

Ultimately, Roddenberry's stance on continuity was that it was whatever he wanted it to be, whether it was broadcast or not. Richard Arnold famously noted that Roddenberry would even decide that he no longer considered bits of TOS canon, and that his stance on what was canon and what was not even changed from day to day, but by that time Gene wasn't in charge anymore.

Star Trek is replete with contradictions and inconsistencies, and we of course it's fun to play the game where we try our best to reconcile these inconsistencies with continuity, logic twisting, speculating on things to fill in the gaps. It's the same game Sherlockians have played for generations (hence Watsonian and Doylist explanations).

But it's only fun until we reach the point where there's really no explanation satisfactory, so we then stop and quietly exile that point, pretending it doesn't exist. And that's just in the broadcast media. When we add in the tons of licensed and unlicensed fiction, it just becomes overwhelming to juggle.

But here's the ugly, unspoken truth. I once formulated this rule a long time ago for Doctor Who, which has an even looser relationship with continuity and a more massive body of licensed work, and worse still has time travel and history changing baked into its very premise: "It's all canon. It's all non-canon." Canon is whatever whoever writes it wants it to be. Nobody's there to really stop them, and any adherence to continuity in fiction is a choice, not a necessity.

You don't get to define canon. I don't get to define canon. We don't even get to define canon for each other. All we can do is define a basis for discussions to take place. It's so easy for you to assert something and me to shoot it down by saying, "oh, that's not canon," without addressing the meat of the argument itself - not whether something is canon or not, but whether that something actually makes sense. And that kind of impasse becomes infinitely more impassable if we don't have a standard definition of canon to begin with.

So questions like, "Is Prodigy still canon," are ultimately non-productive (and a classic closed question) because the answer really is, "It's canon until someone who has the authority and the ability to say otherwise says so, and even then."

And we can pile qualification upon qualification upon this kind of definition until we miss the forest for the ever-expanding numbers of trees, when in the end Star Trek is fiction, fiction is malleable, and the hallucination can only be shared if we share common ground as to what exactly the hallucination is.

So it's not availability rules, it's not Roddenbery rules, it's simply the agreed on rules of engagement that canon is whatever's been broadcast by the main rights holders. Discussion is only productive if we agree on a foundation like this. Otherwise it dissolves into headcanons and no one is more valid than the other. So the conversation trundles to an awkward halt (and sometimes name-calling) as we hug our individual ideas of what constitutes canon and refuse to accept anything otherwise. It's a fun-killer.

And that's why we really shouldn't quibble about canon.

Prodigy is canon insofar as everything is canon when it happens on screen. I like to think of the canon containing all Star Trek works right down to your fan fiction novel or your "head canon" with some things being more canonical than others and with those things which are seen on screen either as a film or TV episode are the most canonical of the canon. Prodigy still fits that bill. I doubt that there will be anything as canonical that would "undo" what Prodigy did. But if that happened we would just incorporate the conflict into the canon like we always do.

I'd say it'll probably remain canon just not discussed often, since even if it was still available it is still a children's show. There might be Easter eggs and references to it though, like how the Ghost from Star Wars Rebels (a kids Star Wars show) shows up in Rogue One a few times but isn't focused on or talked about