Of course it is possible to update beliefs based on evidence if priors are numbers. If I flip a coin but don't show you the outcome, you will rationally assign a prior of 0.5 to the proposition "The coin landed heads-up". If I then show you that the coin did indeed land heads-up (or tails-up) you will update your assessment of the probability of "The coin landed heads up" to something close to 1 (or 0).
Now, it is possible to do Bayesian analysis on probability distributions rather than "bare" priors, but that is just a way of assigning priors to a family of propositions rather than single propositions. For example, "This is a fair coin" is not a proposition to which one can assign a prior because it isn't a well-defined proposition. The intuition behind "fair coin" is something like "the ratio of heads to tails will approach 1 as the number of flips goes to infinity". But no coin can be flipped an infinite number of times, so this is a meaningless definition. Furthermore, consider a coin that always landed on the face opposite that of the previous flip. The ratio of heads to tails for such a bizarre coin would indeed approach 1 as the number of flips approached infinity, but that is not what is generally meant by a "fair coin". So the usual way of dealing with coins is assigning priors to the family of propositions, "If I flip this coin N times the probability that I will get M heads is X" for all possible values of N and M. But each of the propositions in that family gets a prior which is a number.
Now, it is possible to do Bayesian analysis on probability distributions rather than "bare" priors, but that is just a way of assigning priors to a family of propositions rather than single propositions. For example, "This is a fair coin" is not a proposition to which one can assign a prior because it isn't a well-defined proposition. The intuition behind "fair coin" is something like "the ratio of heads to tails will approach 1 as the number of flips goes to infinity". But no coin can be flipped an infinite number of times, so this is a meaningless definition. Furthermore, consider a coin that always landed on the face opposite that of the previous flip. The ratio of heads to tails for such a bizarre coin would indeed approach 1 as the number of flips approached infinity, but that is not what is generally meant by a "fair coin". So the usual way of dealing with coins is assigning priors to the family of propositions, "If I flip this coin N times the probability that I will get M heads is X" for all possible values of N and M. But each of the propositions in that family gets a prior which is a number.