This is basically what's known as survivor bias. I think if you look at history you will find that power shifted to and from faith (and the church) in many waves. But true secularism was not, to my knowledge, seriously suggested for the first thousand years or more of Christian history. In a way this view of theological history suggests that there is a "true meaning" of the Bible that people didn't happen to notice for over a millennium - that the principles of modern society was always there in waiting, waiting for us to understand it. What this Tom Holland might refer to are sayings such as the phrase "Render unto Cesar the things that are Cesar's, and unto God the things that are God's", which does sound like advocating some form of secularism - to modern ears! But of course these verses had many different interpretations over the years.
Personally, I think a much better explanation is that the interpretation of religious texts always happen in a context. Like any text, I don't think it makes sense to claim that there is an interpretation of any religious text that could be called its true meaning, because meaning always arises as a dialogue between the text and its reader, which happens in a certain context, a certain point in material history. You might be able to say that there is the original intention of the authors, but this is not accessible to us. "Render unto Cesar" can perhaps easily be used as an argument for secularism in a Christian context, but I don't believe the phrase can fully explain its inception.
The Bible, like any religious text, contains the blueprint for an infinity of societies. This idea that any part of modern American society, or Hegel's 19th century Prussian society, for that matter, is the natural consequence of what's in the Bible is very strange when you look, both at everything that came before, and also the fact that a very different progression happened in many parts of the Christian world, like Ethiopia or Coptic villages in Egypt. I don't think the theory of "steady progression" holds much water unless you're extremely selective about your facts and perspective.