"Take the proposition that the Earth goes round the Sun. It either does or it doesn’t, so it’s hard to see how we could pick a probability for this statement."
Bayes Factor, the Bayesian alternative to a NHST, is quite a bit different than simply creating the Bayesian equivalent of a t-test. Bayes factor asks "How many times better is my Hypothesis at explaining the data than an alternate Hypothesis". So the Bayesian statement would first pit one model of the Earth's orbit against another. The Bayesian statement of the question of the Earth's orbit would be:
"How much more likely is the astronomical data we've observed to have happened given that the Earth revolves around the sun than it is if the sun revolved around the Earth."
For a more concrete example let's suppose that we have a coin. I think the coin has only heads and you think it is a fair coin, with a 50/50 chance of getting heads or tails. We observe three heads in a row. My hypothesis says that the probability of getting 3 heads in a row given a trick coin is 1. Your hypothesis says that the probability of getting 3 heads in a row given a fair coin is 0.5 x 0.5 x 0.5 = 0.125. My hypothesis explains the data 1/0.125 = 8 times better than your hypothesis. Now suppose the next flip is a tail. The probability of HHHT in my model is 0 and yours is 0.5 x 0.5 x 0.5 x 0.5 = 0.0625. You're hypothesis explains the model infinitely better than mine!
Now we can say that our new Hypothesis is that the coin is fair. Suppose another friend comes along and claims that they thought the coin had a 75% chance of getting heads and only a 25% chance of tails. We flip the coin 5 more times and get HHTTH. Your hypothesis says 0.5^5 = 0.03125, and the friend's says 0.75^3 x 0.25^2 = 0.0263... You're hypothesis explains the data only 1.2 times better than theirs. Clearly, we need more data to feel really confident in one hypothesis over the other.
If you want an even longer example, I wrote a post awhile back about "Bayesian Reasoning in the Twilight Zone" that goes into more detail (including priors)[0]
[0] https://www.countbayesie.com/blog/2016/3/16/bayesian-reasoni...