How do we get hypothesis tests from probability distributions generally?

I was rereading some probabilty and statistics chapters from math books and it looks like the binomial test is based on the binomial distribution and the t-test is based on the student t distribution.

Does it follow that all hypothesis tests are based on a distribution? Or all distributions have a test based on it? There are other probability distributions such as poisson, bernoulli, geometric, uniform…

Is there a generalization that can be made with regards to using any kind of distribution to perform a hypothesis test?