You're right, the second half of your post in effect puts a two-atom prior on "p", with zero everywhere else, and then goes on to use a sequence of such atoms, which would approximate a uniform prior. It's more standard to use smooth priors, because we don't have precise information, but you are right, I was not reading carefully.

You are still in error that there is not a way to describe the uncertainty in your estimate of the posterior probability of system failure. It has a posterior distribution, like everything else in a Bayesian analysis. You would compute it as I described -- Monte Carlo would be easiest.