Today, the idea of covenant is current and constant among Unitarian Universalists leaders, but they're always codes of conduct, lacking the divine referent Puritan covenants had -- and so not really a part of the tradition that's being appealed to. They're also lacking in grace. Covenants, perhaps, but in the sense of making sure Jews don't move into your postwar housing development. (Update. A reader emailed me and thought others may not be familiar with the secular use of the term, in this case a restrictive covenant.) Their appeal surely lies in providing substance after the old Unitarian and Universalist categories were burned to the ground, and for ending argument and reinforcing its appeal to the like-minded. The second I see one of these later-day covenants, I look for the door. (I know there are good people trying to return to a rich theology of the covenant and I wish them well.)
The solemn covenant binds the gathered church apart from the world, though within it. Its role is mainly spiritual, and in our tradition the church in this sense is conventionally tied to the parish or society. (We use the term congregation too, if somewhat improperly, as an alternative to the word church, which itself has at least three meanings: the institution, the spiritual bond and the building. Anyone who claims to be a member of a meeting house had better be a piece of clapboard.) In fact, the parish or society is dominant in both the Unitarian and Universalist traditions and has been for generations. (Do will still teach about the Dedham Decision?) It's that public service -- both in the sense of Sunday worship and the social manifestation of morality -- that distinguishes us. The parish or society is practical and social, too. It won't survey the inner workings of your soul -- which we interpret as freedom -- but it does care the bills are paid. Which is why we have quaint customs like trying to make the collection sacred. And why we invest so much in changing things, especially in society. And why it's hard to look at our history and find a rich traditon of common spiritual practices. Those would belong to the gathered church, which even a century ago may or may not have existed in a particular locale. (This vexed Universalist leaders at the turn of the last century.) Indeed, those member congregations (to use a neutral term) of the UUA that are or were until recently Christian had the marks of that inner church: an annual or oftener communion service, and that most typical of church officers, deacons.
It's been my experience that it's OK for a Unitarian Universalist to have a deep spiritual practice, so long as you were cool about it and didn't come off as a big weirdo. That's parish thinking, and that's fine by me, as long as I can find a place to "go deep" and be that big weirdo. That's the church, in the more narrow sense.
With rentable space and social service nonprofits taking up much of what the parish had to pick up generations ago, I'm prone to go in the other direction: have the church without the parish. Which, in parallel form and different language, is what I suspect a number of neo-Pagan groups are doing. (Let me put a pin in that for reflection, and I'd welcome feedback from members of such groups.)