There are plenty of people who believe that there is a genuine, non-religious 'right wing' to politics; but there isn't. All those people, parties and organisations who self-identify as being on-the-right and against-Leftism but who are not religious are, in fact, on-the-Left.
This is just an objective reality - because to be non-religious is to be of-the-Left. To be anti-Christian is, indeed, the root and motivation of Leftism. Indeed there is no Left and Right, what there is is Left and Religion. By religion I mean specifically Christianity, because we are talking about Western nations and their sphere of influence)
The situation arises because, in the first place, to be not-religious is to restrict consideration to the span of mortal life. This means that the bottom-line is human experience, human emotion, human happiness and suffering. In other words, some version of the (leftist) moral philosophy called utilitarianism.
So, to be non-religious is to agree on the fundamental assumption that what matter most about life is how people feel - and the Left and so-called Right merely differ in methods to affect feelings, and the focus and distribution of feelings.
But I want to draw attention here to the mode of socio-political analysis that is done by the 'non-religious-right' - which is derived from Leftist sociology and politics. This analysis is wholly in terms of one or another Leftist ideology.
For example class and economics (which is Marxism), power (which is anarchism), or status (which is New Leftism/ political correctness).
The non-religious 'right' use Leftist ideology, but just change the valuation of groups. They talk about the ruling Establishment ('Cathedral'/ class/ caste); they divide society into functional groups (peasants, priests, warriors); they use Machiavellian type analysis of power; they discuss the use of education and media to control thought and public discourse...
All these and their basic assumptions, and all other secular schemata, are derived from Leftism. Since the basic assumptions are Leftist, the results will always be Leftist - anything else is excluded by those assumptions.
In a word, the non-religious self-identified 'right' have adopted Leftist metaphysics; that is, their model of how reality is structured derives from Leftism. No matter what they try to do with the tools of Leftism, they will simply loop-back to Leftism.
The difference between the non-religious Left and Right is therefore just 'office politics' among Leftists; a quibble over methods and priorities.
If someone takes the trouble to think about this, rather than simply reacting; they will see that it just is the case. But the non-religious 'right' are not religious for some, very strong and personal, reason - and that reason, apparently, blocks their willingness and/ or ability to think consecutively.
That reason for non-religious anti-Leftists excluding Christianity from consideration is often enough something-to-do-with-sex; or some other personal pleasure that would be excluded by any serious and real religion. But whatever the reason is, it absolutely invalidates their entire project, and indeed dooms them to become unwitting dupes of mainstream Leftism.
Note: Of course, the anti-Left non-religious can easily come-up with excuses for not being a Christian, and the basis that all the mainstream Christian churches are corrupt. However, so are all mainstream institutions of all types corrupt, including political institutions; and that does not deter them from putting all their hopes into political change - so we can recognise that these are indeed just excuses, and not reasons. If they were serious and honest, they would become Christians first*, and then try and work-out how to be real and serious Christians.
*Further note: Of course, it is a deliberate choice to become a Christian. But if someone is genuinely serious about opposing the Left; they need to realise that there is nowhere from-which the Left can be opposed, whether in theory or in practice, except A Religion.