When the path to unity starts with the meeting/church service/ or gathering, then I think its trajectory is skewed from the beginning. I think mission is the better path to unity.
Unite in the furtherance of the gospel. Unite in the suffering and needs of those who aren’t congregating. Unite in the Making of Disciples. Unite in the work of mission.
I suppose there is some discussion needed to decide what that ‘one purpose’ is, but I’d venture to say that it is NOT primarily church services.