When someone makes an argument from ancient authors, I make it a rule to chase up the primary sources to see whether the quote is accurate, and how it fits into the author's thought. Too often they are used to make modern points they would not have recognised.
To reply to a blog post it would be too time-consuming to study all these authors in depth, but let's just take a cursory look through the first one one, Adelard. One needs some basic knowledge of Scholastic theology to see what he takes for granted. For example, although he says his method is to avoid recourse to authority, the preface says his major source was the Timaeus, and though he probably knew Aristotle only secondhand, his dependence is shown in a remark like:
Next, we must take note of their action, in which we must accept not my view but Aristotle's, or rather what is my view because it is his.
So he's not an Enlightenment skeptic, but every inch a scholastic cleric. That's why the work in question takes the form of a socratic dialogue, in which his nephew takes the role of an orthodox, but insufficiently rational, person, as a foil to the Wise Master. Your quote arises from the nephew giving a rubbish argument about the growth of plants in relation to the 4 elements, to which Adelard can respond that it's foolish to invoke God before considering what we know of the rational causes God creates - ie secondary causes - (though of course, what he knew was flat wrong - an object lesson on the provisionality of science in itself).
In that context it can't be taken as setting up any kind of ruling principle of methodological naturalism. And that's entailed by his scholasticism (which surfaces in various of the Qs & As), which following Aristotle sees God as the First efficient cause in every causal chain. So in invoking elemental or humoral explanations, he's saying, "Work up the causal chain before you ask questions about God" - which is an obvious and basic principle of any science, not a call to exclude God in favour of what we would call material causes (but which he calls "efficient causes" in distinction from material, according to good Aristotelian principles.) I've no disagreement with that - and of course it's strictly in the context of plants: he doesn't explore the nature miracles or, indeed, any theology to speak of, in the book, though he does presuppose special Creation and God as First efficient cause.
Some light is cast on this (or his mediaeval mindset, at least, illuminated) by the fact that he had a major interest in astrology and (sympathetic) magic, like most philosophers of his time. Accordingly in one place he argues that the stars have rational souls by their profound influence on the events of the earth, present and future. At one point the governing effect of the moon on the humours is invoked, which is standard mediaeval thinking on the correspondence between the heavenly powers and the earthly humours. But that, of course, implies his acceptance of such a process further up the causal chain mentioned in your quote on plants: plant growth is at least in part the result of intelligent influence from the stars as God's intermediaries in the Great Chain of Being (I mentioned that way up the thread somewhere, didn't I?).
Adelard's efficient causal chain is nonsensical to us (which doesn't change the principle), but would run: First Cause God [as both Creator and Prime Mover] - secondary cause the Stars [as delegated intelligent governing agents] - tertiary cause the elements - quarternary cause the humours - final effect the growth of plants.
The most interesting part to me is that the close of the book shows the limited scope of his book, or perhaps his personal interest - rather than the scope of legitimate enquiry. Nephew says that they've now explored all the way up the causal chain of things, so they now simply have to talk about God, bearing in mind some say there is no God, and so on. Adelard replies that it's a very good question which he will address, and that he finds it easier to say what is not true of God than what is (forsehadowings of Aquinas negative theology there) ... but look at the clock - it's time for bed and he'll talk about all that in the morning. Which of course he never does, either because he sees his role only as a natural philsopher, or because he's a heretic and doesn't want to go there in public.
That's a lot of detail, but you asked my opinion on him, so I sought to understand him in a limited time from his own work. And that makes it clear he's by no means excluding God from the causes in nature, as in semi-deism, still less giving it any kind of autonomy, as in open-process theism. His reasons for avoiding theology are either personal or thematic: he simply doesn't address the kind of questions of divine action covered by those like Aquinas in his teaching on creation, miracles, or providence - though interestingly he does exclude the human soul from natural investigation:
"The discussion of the soul... has little or no relevance to theoreticians or philosophers. since its creation and infusion belong to the divine power in such a way that human reason can have no sensual experience of its disposition and only faith can judge its worth."
My suspicion is that digging into the primary sources for the other writers would uncover similarly nuanced understandings of their quotations, fully compatible with mediaeval understandings of God's immanance in nature - but that legwork is for someone else to do, if they have the energy.