Political and social conservatism is normally associated with (traditional) religious belief, and conservatism without religion can be seen, with some justification, as a slightly anomalous position.
Irving Kristol called it 'thin gruel'. This is one way of looking at it: merely as a watered-down version of conservatism.
But removing, say, Christian beliefs from Western conservatism involves (I suggest) more than just a weakening or watering down. It involves deeper changes - and choices.
For example, does one stay with Christian morality or wipe the slate clean and develop a different morality entirely?
Many non-religious conservatives have looked back to the Greco-Roman world for inspiration, to a time when Western culture was flourishing and had not yet succumbed to Christian doctrines, rituals and ways of thinking. Morally, it was a very different world.
The moral assumptions of the post-Christian world are however, more often than not, Christian assumptions, as the general population in Western countries has arguably internalized large chunks of Christian ethics.
Furthermore, so-called progressive thought is arguably a secular version of one particular aspect of Christian morality. The whole Marxist and broader left-wing tradition is based squarely on certain Christian moral ideals, and could be seen simply to represent an attempt to apply Biblical notions of justice and equality in the here and now - or at least the near future - rather than banking on a supernatural savior.
I think the relatively recent shift of the mainstream churches to the political left can be explained in a similar way. They have in effect ceased believing in a compensatory spiritual realm.
So there is radical and radical. There is standard, left-wing political radicalism which seeks to overthrow the 'unjust' status quo and replace it with a new world order based on (Biblical notions of) justice and equality.
And then there is the deeper radicalism of conservatives who have not only rejected Christian beliefs but Christian morality as well.
But, because most of them have such good manners, and so will refrain from saying what they really think, it will remain largely hidden from view.