I suspect not establishing a state church was a mistake.

Exempting places like China, where attitudes towards the divine have long been fluid and transactional, and belief or disbelief was never much an issue, the most irreligious countries are those in Europe which had established churches. Over time the result has been a nominal allegiance, without much force, whether individual or civic. Religion became regarded as just one more aspect of state bureaucracy, and the general distaste for the latter eroded and engulfed the former. I have little doubt religion would have less political force here and now had the Founders established some variant of the Anglican denominations as the church of the United States.