Look, I have some limited experience with these groups in background and I read a book on the subject, so while I don't claim to be an expert at all, there are some things I've learned:
The term Missionary Baptist dates back to the 1800s as others have save regarding controversy on how church should work. It wasn't just about missions, but rather a clash of styles of church. You don't see as many uses of the term Missionary Baptist because that's essentially the winning side, but it's unfair to totally paint those who are Regular, Primitive, etc as totally against missions. Their way of doing church was suited to a different style of life that clashed with larger denominational trends.
Church in Appalachia (where this centered) is different from church in the south in approach. A good way to learn why and how is the book Appalachian Mountain Religion A History by Deborah Vansau McCauley. She's a little wordy at times, but she does a great job of tracing the lineage.
To very briefly summarize, most of these groups share certain features of worship in common. Footwashing, love feasts and revival culture are the primary things that we might notice at first glance. The Baptist revival tradition was very strong in Virginia and what was then the western part of Virginia (now WV). In addition to this, heavy piestic influences came from Anabaptist traditions as well as some early vestiges of revivalism borne out of Scotland, even prior to the Ulster revival.
This would, in effect, lead to a type of what we now call Charismatic or Pentecostal BUT it would not necessarily be the Asuza Street strand. Thus, you have a wide range of Calvinistic and Arminian beliefs that look very similar in praxis. Interesingly, aside from the Landmarks, a number of these churches will share pulpits with other theological persuasions and possess the same types of culture that can navigate one anothers' services. Revivals are key to the functioning of the church. Sometimes they're called "big meetings" or other terms.
To try and wrap up a lengthy post, key to understanding the dynamic is that the pastors are viewed as on equal footing as members. There are no seminaries and churches are generally quite small and locally-oriented. People in the community can come, but you're often talking about very small communities.
This clashed with larger denominations. The Methodists were coming out of their revivial stage in early America, as were the Baptists. This lead to them looking down upon mountain religion even though it looked a lot like how they used to be in earlier times.
Essential to understanding why this matters is the concept of "getting" religion. People of this tradition view preaching, dancing, and whatever other activities as something where the Holy Spirit figures prominently. Thus, he will guide people into church and put them ultimately where God wants them. This clashes by very nature since it's more of a grass roots push rather than a top-down system which most denominations were either developing or were used to at this point in time. It's overly simplistic to state that these churches just didn't want to field missions.
I'm reaching a little bit here perhaps, but you must understand in their view faith spreads more like a brush fire. Local people nearby would be touched who then go and start other churches down the road. It's almost a farm team system where new pastors and elders are apprenticed without some incorporated body being in charge.
The concept of missions to another area was an outside concept and not exactly on the radar. Whereas the larger denominations viewed the people as generally backwards, poor, ignorant and in need of a home missions movement to fix it all. The fire analogy here is that missionaries would go out and start their own little faith fires in diverse places beyond the horizon.
