I’ve been learning more about the enduring mystery religions of ancient Greece, some of which lasted a thousand years or more. These cults coexisted with the religion of the polis, as well as the household devotions each family practiced. They were different from either one in that they required initiation; Hellenes didn’t otherwise see religion as a choice, any more than they thought they could choose the color of the sky.
Today, the three types of practice are a bit more blurred. Initiations are not uncommon, but don’t always involve the sharing of secrets. (Contrast that with the Eleusinian mysteries, the secrets of which were never written down, and remain unknown.) The same groups which practice those initiations also tend to perform public rituals; in a society where not every citizen is already part of the religion, this makes sense. Honoring household gods such as Zeus Ktesios and Hestia is likewise taught by these groups to the new initiates, since they did not learn those rituals as children.
What we have in this amalgam is really sort of a modified mystery tradition, in that the querent is initiated to learn “secrets” which would have been common knowledge in days of old. But the structure of mystery is not complete; more than initiation is needed for that. Mysteries do involve secrets, and from what we have learned of some of the ancient mystery religions, those secrets sometimes involved significantly different beliefs than those taught publicly. For example, the Orphics were vegetarians who avoided animal sacrifice and believed man to be descended from Dionysus.
There is room for more mystery religions under the umbrella of Hellenismos. I believe that they can only strengthen the religious tradition as a whole.