The point is that there is a decision method that allows me to decide without anyone having to make a prior.

Say, the cost of trial is a, the cost (utility loss) of missing valid cure to strategy failure is b, you do the N trials , N such that a * N < (the probability of trials given assumption of validity of cure) * b , then you proclaim cure not working. Then you can do more trials if the cost of trial falls. You don't know the probability and you still decide in an utility-maximizing manner (on choice of strategy), because you have the estimate on the utility loss that the strategy will incur in general.

edit: clearer. Also I am not claiming it is the best possible method, it isn't, but it's a practical solution that works. You can know the probability that you will end up going uncured if the cure actually works.