> and no other prior knowledge about success probability

This phrase is misleading, as Laplace's Rule of Succession is equivalent to assuming a uniform Bayesian prior over all values of p. That is, before any experiments, a 50% chance of success. Depending on the situation, this may be roughly accurate or wildly wrong. You cannot appeal to this rule to resolve the situation.

Well, obviously if we have a better prior, then that's better. But assuming no other knowledge, and especially if we think that other people's priors could be intentionally misleading, this rule seems to offer the best estimate.