The heart of the Non-Denominational Church, and the United State of America, are found in the Mid-Atlantic, formerly the most urbanized place on Medieval America, and one which tries to hardest to maintain the continuity of infrastructure of the Industrial Era. Courtrooms and hospitals are much more likely to be found under the umbrella of the Non-Denom Church, and as such clerics, physicians, lawyers and even some detectives are considered colleagues to a certain degree. And there are probably even the occasional polymath, who engages in various practices Umberto Eco's In the Name of the Rose, crossed with our image of the Founding Fathers is probably an approximation of what such a person might be.
In some ways, that doctors, justices and priests and priests are part of the same group allows a much smoother engine than that found in some of the more decentralized feudal kingdoms, but it does take away some checks and balances that results in corruption. Conflicts of interest, and maybe even the occasional conspiracy would form. If a legal, medical, and theological expert all agree that a parcel of land is best suited to a particular purpose, who's going to argue?
As for the concept of the Non-Denominational Detective--those are probably fairly rare. Murder and theft usually falls under the realm of secular leaders, and many aristocrats, particularly warlords, do not care about the fates of peasants or sex workers. They barely lift a finger over livestock theft, and are quick to round up and execute the usual suspects to make an example. But members of the Church are amongst the most learned of the Medieval World, and they are often consulted. And if a commoner can scrape enough silver, or make a moving emotional plea, maybe a cleric might be bothered to look into a crime. After all, their brains have been wired to as questions and look for answers. And what better way to get converts than be the ones to provide something rare in these times--justice?
As for the concept of the Non-Denominational Detective--those are probably fairly rare. Murder and theft usually falls under the realm of secular leaders, and many aristocrats, particularly warlords, do not care about the fates of peasants or sex workers. They barely lift a finger over livestock theft, and are quick to round up and execute the usual suspects to make an example. But members of the Church are amongst the most learned of the Medieval World, and they are often consulted. And if a commoner can scrape enough silver, or make a moving emotional plea, maybe a cleric might be bothered to look into a crime. After all, their brains have been wired to as questions and look for answers. And what better way to get converts than be the ones to provide something rare in these times--justice?