I won't write an extensive discussion of this here, as I have done so previous on askhistorians (here and here) and I would be covering much of the same material. However, I will briefly address the specific points you bring up.

So I think it is worth addressing the way this question has been framed from the start. Historians don't tend to inquire as to whether Christianity was essential to the rise of science, as there is no good reason to suppose that something like modern science couldn't have developed in a non-christian context. This would require a great deal of counterfactual argumentation of the sort that historians don't tend to be interested in. Furthermore, the suggestion is itself highly implausible, given that there is nothing about Christianity per se that was essential to the rise of science. However, it happens that modern science as we understanding it developed in close relationship with latin Christianity in the medieval west.

On the general quotation you note. It is true that Christianity conceives of God as rational etc., but taht statement is either so general as to be meaningless or is not sufficiently general as to cover the breadth of Christian thought. In the former sense, it is sort of meaningless as we see equivalent conceptions of God (or something like that) already in pagan monotheism hundreds of years before the advent of Christianity, going back to the Pre-Socratics and seen clearly and systematically in things like Plato's demiurge and Aristotle's prime mover. So in this sense it is not obvious that Christianity is, as such, bringing anything new to the table. On the other hand, given that Christianity is a revealed religion, there is an important anti-rationalist bent within Christian thought, one that is not adequately captured by any more specific reading of that description. For example, from Cosmas Indicopleustes's Christian Topography:

And when they find the Old and New Testaments to be in mutual harmony, they abide therein firmly grounded and immovable, in nothing confounded by their adversaries. But those on the other hand who prank themselves out in the wisdom of this world, and are self-confident that by scholastic reasonings they can comprehend its figure and position, scoff at all divine scripture as a mass of fables, stigmatising Moses and the prophets, the Lord Christ and the Apostles as idle babblers, and given over to vain delusions; while with supercilious airs, as if they far surpassed in wisdom the rest of mankind, they attribute to the heavens a spherical figure and a circular motion, and by geometrical methods and calculations applied to the heavenly bodies, as well as by the abuse of words and by worldly craft, endeavour to grasp the position and figure of the world by means of the solar and lunar eclipses, leading others into error while they are in error themselves in maintaining that such phenomena could not present themselves if the figure was other than spherical.

So this sort of general statement doesn't do much good in grounding our understanding of the role of Christianity in the history of science.

Now it is true that that sort of quotation can more accurately describe Christianity as it developed in the latin tradition, particularly by the end of the 12th century. (Take for example the normally cited image, in an early 13th century Bible moralisée, of God as architect of the world.) But, of course, it seems a bit strange to me to pin this sort of rationalism on Christianity as such, as this sort of Christianity developed out of a particular moment that saw an increased desire for knowledge of the world, knowledge that was explicitly driven by the reimportation of classical thought, particularly in the translation of Aristotle. So again, this is not obviously about Christianity per se, but Christianity as it developed at a particular historical moment.

But anyways, as I noted above, Christianity is normally taken to be a major (though contingent) factor in the development of modern science. However, there is no real consensus on the way in which it was uniquely important to this development. Just to provide three examples, David Lindberg seems to argue that the most important factor was the long standing financial and institutional support of science that the Church in the West provided; Amos Funkenstein develops a fairly complex argument to the extent that, as intellectual systems, there was an increased convergence between theology and science from the thirteenth to seventeenth centuries; and Edward Grant argues that the relationship was more contingent, with the intellectual environment of the medieval university fostered a culture of inquiring into the world and that particular events, like the condemnation of 1277, fostered a crucial break with Aristoteleanism allowing, very slowly, for the development from "empiricism without observation" to the experimental sciences as they emerged in the scientific revolution.