It's an Anglo-Saxon thing derived from English common law, so it's not obviously relevant in other parts of the West, continental Europe for example, though it's not unknown there.
Then it's true that something like Blackstone's Ratio has been around in Europe for a long time, but I'm not sure how useful the ancient history is. Once Blackstone used it, it took off in a big way. So some kind of answer is needed to the question of why it became so important then.
I don't think it has much to do with Christianity. The function of the dictum is to legitimise various policies in the criminal justice system which are highly consequential in terms of who is convicted and who is acquitted. Prior to Blackstone, Christianity had an alternative and highly developed approach to this, exemplified by the idea of a half-proof and most maturely expressed in the teachings of Baldus - see Franklin.
It has been used historically like a passage from the Bible, as something that supplies justification rather than something that seeks justification, as you would have it do. This started with Blackstone himself, who provided no argument to support the principle.
It is very hard to say how central the idea is today. Historically, it has been and many policies were justified explicitly by recourse to the principle. But today the policies are largely entrenched and all sorts of justifications are offered for them. A kind of scholarly crucible for this discussion is the question of the standard of proof - "beyond reasonable doubt" or "beyond any doubt" or "on the balance of probabilities" etc. Here ambition in the philosophy of law is largely limited to the ahistorical goal of justifying the existing arrangements, and even this remains elusive - see Picinali.
It is certainly possible to argue that the price society is paying (not being harsh enough on the accused) outweighs the benefits gained. For about 15 years Larry Laudan in the United States has been trying to make this argument, partly through the use of empirical evidence.
The presumption of innocence is a bit like Blackstone's Ratio in that it is not a policy in itself; rather it is a principle that can be used to justify a policy, such as excluding a certain kind of evidence. So compatibility per se is not really an issue in my view.
I think the best starting point for thinking about Blackstone's Ratio is to note that it assumes an objective frame of reference in which defendants are truly innocent or guilty and the court may get its verdict right or wrong. The dictum then seems to express a valid value judgement that ought to be consequential and in this way I think Blackstone promised a basis for policies that seemed quantitative, objective and democratic – in a word, modern. No wonder it was popular.
But Blackstone failed explain how to make the value judgement consequential. It isn’t clear how one gets from the ratio of 10:1 to any particular policy, as opposed to one slightly stronger or weaker, or indeed much stronger or weaker. Equally, it isn’t clear what policy options and practical consequences we should be comparing if you advocate 10:1 and I advocate 100:1. There is a methodological gap. Thus Blackstone’s Ratio left the law in an awkward state, half in and half out of modernity. Subsequent generations have been unable to fix the problem and gradually the lack of a cogent argument has I think caught up with the dictum, which has become less central.
You can find more about this, including my attempt to fix the methodological gap at Blackstone's Ratio Revisited.