I don't think the two are so analogous. A problem with placing a high value on finding exact closed forms for integrations is that it encourages behaviour like

I don't think the two are so analogous. A problem with placing a high value on finding exact closed forms for integrations is that it encourages behaviour like the drunk looking across the street from where he lost his key because that is where the streetlight is. Most integrations just don't have closed solutions and we have to live with that fact. Bayesian statistics has been liberated by the ability to perform many dimensional integrations on the kind of likelihood functions appropriate for the problem, where before the advent of modern computational techniques Bayesian statistics had a reputation for concentrating on artificial problems that we happened to know how to solve.