Re: the “p-less-than” and “p-equals” I don’t discuss it because “p-equals” is just “Bayesian statistics” — that is, conditioning on the observed test statistic. Special names aren’t needed for different conditioning events. The only time you’d condition on the test statistic being part of a set (“p-less-than”) is when you didn’t know the exact value (can happen with some reporting practices), or for simple pedagogical reasons. You condition on what you know to make an inference.

Re: risk versus rate, it sounds to me like you’ve moved from a frequentist interpretation (rate) to a Bayesian one (in terms of a posterior probability of this particular hypothesis). That’s ok, but then you’re affected by a similar problem (the reference class problem) from a different perspective. You need to specify the prior probability of that specific hypothesis. You can’t appeal to other hypotheses unless you’ve specified both 1) the probabilistic relationship between those hypotheses being true and your particular hypothesis being true, and 2) the relationship between the observations about those other hypotheses and their truth value (that is, the Bayesian analysis is joint across all the hypotheses, and you can then marginalize over the other hypotheses).

A Bayesian approach only gets you out of this problem if you are able to specify a subjective prior that is specific to whatever problem you’re testing. But then you’re open to the charge that the answer is arbitrary. (See https://philpapers.org/rec/HJETRC)