Yeah, Neyman’s proof of Laplace’s version of the rule of succession is nice. The reason I think this kind of approach can’t give the full strength of the conjugate prior approach is that I think there’s a kind of “irreducible complexity” to computing Beta(α,β) for non-integer values of α,β. The only easy proof I know goes through the connection to the gamma function. If you stick only to integer values there are easier ways of doing the computation, and the linearity of expectation argument given by Neyman is one way to do it.
One concrete example of the rule being used in practice I can think of right now is this comment by SimonM on Metaculus.