I think that problem is in the sentence
You should expect that, on average, a test will leave your beliefs unchanged.
That happens to be not true. A test which ouputs useful information WILL change your beliefs. Especially given point 2, one can say "Any informative test will always change your beliefs".
What's tricky here is expectation. You expect your beliefs to change but you don't know in which direction. So your expectation is for zero change even though you know that you'll get some non-zero change.
This looks paradoxical, but is the entirely standard way in which statistics (in particular random variables) operate. Consider a toss of a fair coin. The expectation is half heads half tails which is guaranteed not to happen. You know you'll get either heads or tail but not which one of those two. The expectation will not match the outcome -- all it can do is be equidistant (appropriately weighted) from all possible outcomes.
Recently, I completed my first systematic read-through of the sequences. One of the biggest effects this had on me was considerably warming my attitude towards Bayesianism. Not long ago, if you'd asked me my opinion of Bayesianism, I'd probably have said something like, "Bayes' theorem is all well and good when you know what numbers to plug in, but all too often you don't."
Now I realize that that objection is based on a misunderstanding of Bayesianism, or at least Bayesianism-as-advocated-by-Eliezer-Yudkowsky. "When (Not) To Use Probabilities" is all about this issue, but a cleaner expression of Eliezer's true view may be this quote from "Beautiful Probability":
The practical upshot of seeing Bayesianism as an ideal to be approximated, I think, is this: you should avoid engaging in any reasoning that's demonstrably nonsensical in Bayesian terms. Furthermore, Bayesian reasoning can be fruitfully mined for heuristics that are useful in the real world. That's an idea that actually has real-world applications for human beings, hence the title of this post, "Bayesianism for Humans."
Here's my attempt to make an initial list of more directly applicable corollaries to Bayesianism. Many of these corollaries are non-obvious, yet eminently sensible once you think about them, which I think makes for a far better argument for Bayesianism than Dutch Book-type arguments with little real-world relevance. Most (but not all) of the links are to posts within the sequences, which hopefully will allow this post to double as a decent introductory guide to the parts of the sequences that explain Bayesianism.