But I am willing to bet VERY FEW if any politicians over the years have
trundled to the chaplain to discuss the ethical underpinnings of their
votes, or what they should do about X or Y policy.
It would certainly be interesting to find the answer to this, but it's unlikely to be possible.
I think Madison had it right, but mostly I think it doesn't make much difference in this case. Unlike, say, abortion or contraception policies--which may be influenced by religion--opening prayers in Congress just don't affect government's impact on people's lives. If they had a strict rotation to include all faiths (and an occasional poet), it would probably still have no impact.
A strict separation of church and state seems like an idea worth defending, but one should be pragmatic about it. The Church of England is a state church, but it doesn't affect governance there, and religious freedom is unfettered. In the US, government policies for education, healthcare and civil rights are all being influenced by religious factions, which seems more worrisome than a ritual prayer in Congress. Best to choose one's battles.