Of those Churches what is their form of governance that allows them to run smoothly lacking fulltime ecclesiastical professionals? How does one ascend into a leadership role in the Church?
Quakers and Mormons have no paid staff in their local assemblies. In the national bodies, however, there is usually some renumeration or support, in order to retain staff. The truth is that very few would consider it wise to have a national organization without any accountability, which is what an all-volunteer lobby would be.
Additionally, many more Baptists than you might think have volunteer staff. Finally, there are a great number of unpaid missionaries, itinerant preachers, and supporters of various causes that may fit into this category. Usually the local pastor is supported by the congregation, but the national staff has many more volunteers.
Finally, if you look at what a Roman Catholic priest earns, it's pretty close to nothing.
All that said, most mainstream Protestant churches will pay their clergy, based on Paul's commentary on paying preachers in 1 Timothy 5:18. Those churches relying on unpaid staff tend to be less mainstream and more "feeling" driven, with the feeling being that preaching is a gift like any other.
Coming from a Plymouth Brethren background, I can speak to the experience of growing up in a church movement which eschewed ordaining ministers and employing them as fulltime servants within local churches. Wikipedia has a decent overview of the Plymouth Brethren, a movement which began in the late 1820s in Dublin, Ireland and quickly spread to Plymouth, England in 1831.
Darby, by the way, has his own translation of the Bible as he was quite adept in biblical languages.
The Wikipedia article goes on to say,
When I was growing up, the Plymouth Brethren (PBs for short) did support missionaries both home (in the U.S.) and abroad, but those who served in the U.S. and Canada were called "fulltime workers." This meant that they were supported financially by one or more "assemblies" (the PB name for local fellowships around the world). For a number of years, the assemblies also had a mission organization stateside called Christian Missions in Many Lands (CMML), which published a magazine devoted to the work of PB missionaries throughout the world.
Another ministry of the PBs was, and still is, Emmaus Bible College, located in Dubuque, Iowa, which, in cooperation with other local colleges, grants four-year degrees in a number of disciplines and majors, including, of course, biblical studies. I attended the school for two years back when it was located in Oak Park, Illinois.
The PBs had as well a financial institution called the "Stewards Foundation" which helped to finance the planting of new assemblies by providing seed money for building a permanent house of worship, commonly called (in years past) chapels. I myself was brought up in Congress Avenue Bible Chapel in Rochester, New York, where I became a lay preacher and teacher for a number of years. Since I moved to Pittsburgh many years ago, the chapel building in which I grew up was sold and the members built and moved into a new facility in suburban Rochester. They called it Cornerstone Bible Chapel. It is doing very well, thanks to committed and Spirit-filled members, elders, and deacons who go above and beyond the call of duty to help grow the church spiritually.
In conclusion, the PB assembly movement has been on the decline for a number of years now, for a number of reasons. We needn't consider those reasons at this time. A new trend has emerged in the last few decades, however, and that is the financial support of elders. The biblical basis for this is 1 Timothy 5:17 NIV:
My own personal feeling (and I could be quite wrong) is that the spiritual vigor of the general membership in many PB assemblies has declined over the past 50 years or so. Worldliness in the form of materialism and a "let George do it" mentality have sapped the strength of many assemblies. The level of commitment that is required for a volunteers-only church to run at least at near-peak level has not changed in the last half-century, but that level has not been reached of late, particularly in those churches which lack a critical mass of unpaid but committed volunteers and leaders.
Paid fulltime elders and even paid staff members have helped to bring new vigor to some assemblies, but if a pervasive worldliness takes over in any given assembly, particularly to the point where there are simply not enough volunteers to bear the burdens of leadership and followership, the typical distinctives long associated with this once vital movement are likely to fade into oblivion.