my reasoning about "consecutive coin flips"

Yes, and note that this part -- "that I have to start considering that the die is loaded" -- is key.

but I don't quite see how to generalize that observation

Um, directly? All models which you are considering are much simpler than the real world. The relevant maxim is "All models are wrong, but some are useful".

I think you got caught in the trap of "but I can't change my prior because priors are not supposed to be changed". That's not exactly true. You can and (given sufficient evidence) should be willing to discard your entire model and the prior with it. Priors only make sense within a specified set of hypotheses. If your set of hypotheses changes, the old prior goes out of the window.

The naive Bayes approach sweeps a lot of complexity under the rug (e.g. hypotheses selection) which will bite you in the ass given the slightest opportunity.

Sometimes the alternative hypothesis (e.g. the breakages are connected) is not apparent or obvious

Yeah, well, welcome to the real world :-/

My wife seems to think that making explicit model-based predictions in the first place is the problem.

She is correct if your models are wrong. Getting right models is hard and you should not assume that the first model you came up with is going to be sufficiently correct to be useful.

System 2 really shouldn't actively lead me astray.

I see absolutely no basis for this belief. To misquote someone from memory: "Logic is just a way of making errors with confidence" :-P