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Alice: I'm hungry.
Bob: Want a sandwich?
Alice: Meh... not really.
Bob: How about a hot dog?
Alice: Oh, yeah, that sounds good.
Bob: Aha! So you do want a sandwich.
Alice: What?
Bob: Well, technically a hot dog *is* a sandwich, so when I asked you if you wanted a sandwich and you said no, you were mistaken.
Alice: ...
Bob: (complicated internet argument proving a hot dog is a sandwich).
Alice: Okay, fine, whatever. I'm hungry. Just make me a "sandwich".
Bob: (hands Alice a ham and cheese sandwich)
Alice: ...okay, see, this is literally the exact thing I *didn't* want.
Bob: But you said you wanted a sandwich!
What's happening in this parable is that Alice is making an honest attempt to communicate her preferences to meet a need, and Bob is playing word games. The content or strength of (complicated internet argument proving a hot dog is a sandwich) is completely irrelevant to that assessment: whether or not Bob is justified in conflating "hot dog" with "sandwich", it is indisputable that Alice has clearly communicated a preference and Bob has provided something which does not satisfy that preference.
The "taxation is theft" argument is the same. People and societies throughout history have established legal regimes which codify taxation into law but ban theft, which clearly communicates a preference for taxation and a dispreference for theft. There is nothing remotely confusing or ambiguous about this distinction, and so just as any person operating in good faith can instantly distinguish between a hot dog and a sandwich, any person operating in good faith can instantly distinguish between taxation and theft. Regardless of the content of (argument that taxation is theft), any attempt to substitute one for the other is justifiably going to be met with confusion and annoyance by people who are trying to use words in their regular senses to communicate regular pragmatic concepts in order to meet their needs.
I submit that deliberately making it harder to communicate detracts from reasonable discourse and should be avoided, as a general rule.
In general, there is no requirement that any particular word must describe a coherent category with clear membership. This seems like an inherently different concept for people to master and internalize, because examples keep popping up throughout history - from Diogenes mocking Plato's "featherless biped" to "is cereal soup", we seem to always want to abstract a category definition from a collection of known category members, and then try to expand the category by re-analyzing non-members using the new abstracted definition. Sometimes this works and yields insight, but sometimes it produces strife and confusion. The key is not to mistake the proposed category definition for an ontological fact about the world, but rather use it as an experimental mental model which might or might not yield insights.
In the case of the "taxation is theft" argument, if the insight is something like "don't fund an art museum with taxpayer funds because taking someone's money to buy art is unjustified" I'm not even sure that "taxation is theft" is doing any intellectual work at all other than bringing a loaded term into the conversation so that people focus on arguing about rhetoric rather than arguing about the opportunity costs of art museums and the consequences of a society with a higher or lower tax rate. In other words, you definitely could have realized those are important questions to consider without first conflating taxation and theft. I think that in this case, as with Bob's ham and cheese sandwich, the conflation is just making it harder to communicate about what we actually want as a society.
To the extent that the non-central fallacy represents an instance of trying to reanalyze group membership based on some insight about what group members have in common, it can be a useful tool for producing hypotheses. But to the extent that it is used as a rhetorical device, it seems to make it harder to communicate effectively and harder to think clearly about the object question, and seems to produce outcomes in conversations that are very similar to the outcomes from bad faith arguments.
Teacher here: you're conflating course content with assessment, which is a mistake - the two are logically independent.
I assume the purpose of a course called "Survey of the Arts" is to get you to learn some content - that is, some set of facts about art movements. Because it's a survey course it should be broad general knowledge. Survey courses often exist to allow students to dabble enough in a subject to see if they're interested in any particular field. I did a linguistics survey class and discovered I was really interested in sociolinguistics and not very interested in phonetics. That in itself was useful.
The purpose of an assessment is to measure how much content you learned. A common assessment strategy is to try to take a sample of your knowledge. A paper can be a good sample of your knowledge even if you use your knowledge to make a bad argument - the Survey of the Arts class might not consider "making a good argument" to be a course goal.
Also, even if an assessment is bad, or if the task is assessed poorly, that doesn't make the knowledge you gained from the course useless.
I'm a teacher and in the country where I live we had a few weeks of school in-person this September-October until the fall wave got too high and we all went home. School was run with a reduced class size and distancing rules in place, but try telling that to a sixth grader. At one point a kid jumped out of his chair and ran up to me and I had to tell him to back up and go back to his seat. Another kid asked, "Mr. Z, are you scared of coronavirus?"
What did that feel like? It felt like being offended. I sort of mentally seized up, and my chest muscles clenched. I was able, with some effort, to acknowledge that yes, coronavirus was a concern for me. I've thought about that moment a lot.
My theory at the time was that we're conditioned to avoid admitting to being afraid of something, especially publicly and on short notice. At least when I was growing up, if I admitted fear, I'd generally be mocked for it, so I learned not to do it. And when someone asks me if I'm afraid - or accuses me of being afraid - I get angry at them. I want to tell them, "no, I'm not 'afraid' - I'm rationally concerned about bad outcomes." I think in part the accusation of fear angers me because it implies that I am somehow being irrational, or letting my emotions get the best of me - whereas my belief is that if I'm "afraid" that means that there is real danger which is dangerous and likely enough to warrant some action to mitigate the danger. But in general, if someone asks me if I'm afraid, what it feels like is they're accusing me of being irrational. So my impulse is to deny it, or to try to "rationalize" the fear.
But I think this is a destructive impulse - I think being afraid to admit you're afraid leads people towards collective stupidity, like reopening schools before it is safe to do so. Speaking of which...
This Monday we're supposed to return to in-person instruction, despite cases being almost as high now as they were when we closed. Parents are just tired of their kids, I guess. Anyway, because I have a specific risk factor, and live in a multi-generational household, I asked my principal if I could continue teaching online and just have them project me into the classroom for the seven kids at a time that will be allowed in the room in person. I ended up speaking with the director of the school yesterday and she told me I sounded scared. Without hesitation I agreed that I am indeed scared. But being able to smoothly admit that seems to have taken, let's say, many hours over many months of introspection and reflection and mental preparation and specific resolve. Basically I had to have so much confidence that my fear was justified and that mitigating the danger was necessary that I felt like I could absorb the hit to my social status or reputation that would come from being a person who admits to being afraid. Or at least, that's what it looks like from the inside. I know that I did the introspection and preparation, and I know that my physiological and mental reaction to being asked if I was afraid of coronavirus changed between October and February, but I obviously can't prove that the introspection and preparation caused the change.