I'll have to pick up Voegelin work sometime, but I've seen a mostly welcoming response from historians regarding Israel's work. Of course, there's bound to be criticism here and there. It's curious that your philosophy teacher found Israel's use of "Radical" a pet-peeve, given that the original Greek meaning isn't much in play nowadays, whereas radical is understood to mean revolutionary. But even so, Israel dedicates a few chapters on the "Recovery of Greek Thought"; how proponents of the radical Enlightenment were influenced by ancient atomists, materialists, and atheists such as Lucretius, Epicurus, Strato, among others. So in a (stretched) sense, the Greek etymology is viable here, too.
My point is that a philosopher working during the Enlightenment could have one foot firmly in radical thought pertaining to certain issues, while the other foot is set firmly in moderate thought on others, which Israel himself acknowledges. I haven't read
Revolution of The Mind, but I think, in the quote you offered, Israel is merely generalizing. I think it is very safe to say, from what I've read in his other work, that believing in a providential universe, by itself, does not necessarily entail a commitment to monarchism contra democracy. To some extent, you are right that for the radical Enlightenment philosophers, a particular metaphysics provides a consistent basis, or extension towards an egalitarian and democratic socio-political society, and I believe Israel does mention that somewhere. So, as you said, it is a "key factor", to an extent. But, as with Hume, (and Pierre Bayle, if I recall correctly), among a few other lesser names, that's not always the case.
It should also be noted that while the French Revolution had leaders influenced by the radical philosophers, other, more influential leaders, and who ultimately grabbed the reins of the revolution, such as Marat and Robespierre, were influenced by "moderate" enlightenment thinkers such as Rousseau and Montesquieu, not by Diderot et al.