The answer depends entirely on how you define "the mysteries".
I am more than humble enough to admit that there are lots of folks who are more "enlightened" than I will ever be, and most of them are not Freemasons, so clearly we can not claim to have a lock on "enlightenment". Nor do we have an exclusive on "morality" (if that's part of your definition of "the mysteries"). Indeed, one often cited description of Freemasonry is "... a peculiar system of morality, veiled in allegory and illustrated by symbols." Note that it's "a" system, not "the" system. Again, better minds than mine have long propounded the same moral truths that we discover through our labors in the Craft, so clearly there are many paths to such knowledge.
What the non-Mason can not have is the particular experience of gaining enlightenment via our "peculiar" allegories and symbols. I believe that it can be argued that the journey is part of the deal, no matter what path you choose. You definitely don't get the same thing from reading a book (or web site) that you get in Lodge. One of those "better minds than mine" once told me that what you get out of Freemasonry is directly proportional to what you put into it. It's kind of tough to put anything back into a book, but the opportunities to contribute to Freemasonry's goals are many. For some, it's philanthropy, for others it's ritual work, or maybe just hands-on labor - maintaining the Lodge building or preparing the meal on meeting nights. The point is that you need to be engaged in the give and take to get the stuff that really matters. It matters not if it's Freemasonry or some other "mystery school" - you can't just read about it. You have to live it.