This is an overall well-reasoned post. I don't want the conclusion to be true, but that is no reason to downvote.
I noticed this years ago when the variations of the show Big Brother were on TV in various countries. The show consists of compilations of real people spontaneously talking to each other throughout the day. The difference between this and the scripted conversations we saw on TV before Big Brother is huge. Real people apparently hardly talk in complete sentences, which is why scripted conversations are immediately recognizable as being fake. It's also strange that this is hardly noticeable in real life when you are actually having conversations.
I think one issue with the "person+time" context is that we may assume that once I know the time, I must know whether it is Friday or not. A more accurate assessment would be to say that an indexical proposition corresponds to a set of possible worlds together with a person moment, i.e. a complete mental state. The person moment replaces the "person + time" context. This makes it clear that "It's Friday" is true in some possible worlds and false in others, depending on whether my person moment (my current mental state, including all the evidence I have from perception etc) is spatio-temporally located at a Friday in that possible world. This also makes intuitive sense, since I know my current mental state but that alone is not necessarily sufficient to determine the time of week, and I could be mistaken about whether it's Friday or not.
A different case is "I am here now" or the classic "I exist". Which would be true for any person moment and any possible world where that person moment exists. These are "synthetic a priori" propositions. Their truth can be ascertained from introspection alone ("a priori"), but they are "synthetic" rather than "analytic", since they aren't true in every possible world, i.e. in worlds were the person moment doesn't exist. At least "I exist" is false at worlds where the associated person moment doesn't exist, and arguably also "I am here now".
Yet another variation would be "I'm hungry", "I have a headache", "I have the visual impression of a rose", "I'm thinking about X". These only state something about aspects of an internal state, so their truth value only depends on the person moment, not on what the world is like apart from it. So a proposition of this sort is either true in all possible worlds where that person moment exists, or false in all possible worlds where that person moment exists (depending on whether the sensation of hungriness etc is part of the person moment or not). Though I'm not sure which truth value they should be assigned in possible worlds where the person moment doesn't exist. If "I'm thinking of a rose" is false when I don't exist, is "I'm not thinking of a rose" also false when I don't exist? Both presuppose that I exist. To avoid contradictions, this would apparently require a three-valued logic, with a third truth value for propositions like that in case the associated person moment doesn't exist.
And what would this look like? Can you reframe the original argument accordingly?
I meant leogao's argument above.
It seems the updating rule doesn't tell you anything about the original argument even when you view information about reference classes as evidence rather than as a method of assigning prior probabilities to hypotheses. Or does it? Can you rephrase the argument in a proper Bayesian way such that it becomes clearer? Note that how strongly some evidence confirms or disconfirms a hypothesis also depends on a prior.
a prior should be over all valid explanations of the prior evidence.
... but that still leaves the problem of which prior distribution should be used.
It seems you are having in mind something like inference to the best explanation here. Bayesian updating, on the other hand, does need a prior distribution, and the question of which prior distribution to use cannot be waved away when there is a disagreement on how to update. In fact, that's one of the main problems of Bayesian updating, and the reason why it is often not used in arguments.
X because Y implies X and Y, though not the other way round.