Massimo Pigliucci has been reading Bertrand Russell's Mysticism and Logic. Like a lot of people who come to it for the first time, I think he is reading a few things into Russell's argument that really aren't there, because Russell is hunting much bigger fish than the little fry that Pigliucci has in mind.
Mysticism and Logic basically consists of very early work by Russell; it was published in 1918, but much of it is quite a bit earlier than that, since it's an anthology of essays written between 1901 and 1914, with the titular essay being written in 1914. This is actually pretty important for understanding the work: it consists of the early championship of analytic philosophy of mathematics and the advocacy of a generalized form of the approach used in that field for investigating the problems of philosophy generally; and this was being done in the face of the dominant philosophical approaches of the time. The mysticism that Russell is addressing is not what we would ordinarily think of as mysticism: it's a large family of philosophical positions. The basic idea in the essay "Mysticism and Logic" is essentially a thesis in what we might call psychological history of philosophy; it posits two broad impulses, the mystical and the scientific, that have shaped the course of philosophy itself. Russell is explicit that both are actually necessary for good philosophy; but he thinks that the mystical impulse has too often been allowed to run unchecked. And although he doesn't explicitly say so, it seems clear enough that what he chiefly has in his sights is British Idealism, and is arguing against it by proxy; all four of the theses he primarily focuses on can be found in at least some British Idealists:
(1) "belief in the possibility of a way of knowledge which may be called revelation or insight or intuition, as contrasted with sense, reason, and analysis, which are regarded as blind guides leading to the morass of illusion";
(2) "belief in unity, and its refusal to admit opposition or division anywhere";
(3) "denial of the reality of Time";
(4) "belief that all evil is mere appearance, an illusion produced by the divisions and oppositions of the analytic intellect".
If you want to know where to find people who have this sort of "mystical metaphysics" in spades you look to Bradley and McTaggart and Bosanquet. (Bradley and McTaggart aren't mentioned by Russell in this work, but Bosanquet is referenced in another essay in the collection on the same subject.) But, of course, one of the noticeable things is that neither Bradley nor McTaggart nor Bosanquet fit any of Pigliucci's descriptions of mysticism. They do not "steadfastly refuse to acknowledge any useful role for logical analysis"; they are filled with arguments that involve a considerable amount of logical analysis. They do not hold that "they (and only they) have access to a deeper level of reality, a type of access that is not available to most of us"; they argue that positions opposed to their own are incoherent and that anyone who takes the trouble to follow their arguments can come to recognize this, and they appeal to no faculty or capacity that they do not take to be shared by every rational person. And while Bradley, McTaggart, and Bosanquet are the creme-de-la-creme of British Idealists, it's not as if they are unique in this way. And the people Russell explicitly mentions in the essay as going too far in the direction of a mystical metaphysics, people like Plato and Spinoza and Hegel and Bergson, do not fit Pigliucci's profile, either. When Russell talks about mysticism, he is talking about something that he thinks is false because it consists of answers that have been rigged from the get-go by emotion (in effect, allowing an essential impulse of philosophy, suitable for motivation and good for raising questions, to do what it is not good at, answering the questions it raises), but he is talking about something that in its development is very, very rationally argued. From his perspective it's philosophy that has taken a wrong turn, to such an extent that we should from now on be sharply suspicious of the impulse that drives it when it does any more than spark interest in the world, but it is undeniably philosophical.
Now, one can perhaps hold that Deepak Chopra (to use Pigliucci's example) is likewise an instance of the mystical impulse run unchecked; but taking Russell to be aiming at people of this sort is to take him to be stomping on ants, when he's actually trying to bring down giants. Perhaps there is some fundamental commonality between Deepak Chopra and Spinoza; one is inclined to be skeptical of the association and doubt that there are any more than verbal similarities, but perhaps. They are, after all, both human and perhaps there is some common impulse that they share in developing their views. But Russell isn't attacking (say) table-turning and spiritualism; he is attacking major, and powerfully argued, philosophical systems, trying to diagnose where they all go wrong despite the brilliance used to argue them. And 'argue' is an important word here. Nobody can say that Plato doesn't argue for the reality of The Good, even if they think the arguments fail; nobody can say that McTaggart doesn't have arguments that time is not real; nobody can say that Bergson doesn't give reasons to think that we really do have intuition as well as intellect; nobody can say that Spinoza goes around making claims without arguments. People have said that Hegel doesn't argue, but this comes from the fact, I think, that Hegel likes to present as arguments as if they were just obvious descriptions of fundamental reality; when you look at how he interacts with, say, Kant, it becomes pretty clear that the Phenomenology and Hegel's other works in fact argue for Hegel's main points at very great length. Russell has big ambitions in these little essays.