There Many kinds of church assemblies calling themselves "Baptists," with all kinds of organizational rules, doctrines, constitutions, etc. that set them apart from each other. There is only one single thing upon which they all agree, which is that water baptism is by immersion of the whole person.
However, there are some other characteristics upon which most assemblies are in agreement. These are known as Baptist "distinctives," each of which can be represented by the letters B, A, P, T, I, S, T.
Of these, you will find that one T represents "Two Offices"; that is, church polity is that one person is the Elder. He (Scripturally only a man) is also the Bishop, and is also titled "Pastor."
The other office is "Deacon," and there may be as many deacons as needed and as defined by the church constitution.
So, in most cases, your statement is not quite right. In a Baptist church, the pastor (shepherd) is considered to be both the ruling elder and the teaching elder, as well as the spiritual elder.
Deacons usually tend to the finances, support of needy members, the physical plant, and visitation.
Generally, upon vacancy of the pastor/relder/bishop's office, the calling of a pastor is determined by the vote of the congregation. Once elected by the congregation, the pastor assumes total (and usually unquestioned) responsibility for the church and its conduct. Scripturally, the pastor does not serve at the will of the deacon. Rather, the deacons serve at the rule and direction of the pastor. This is not always the case, but is generally true.
Another distinctive worthy of note here is represented by the letter "A" -- a Baptist church is Autonomous -- its next higher authority is The Lord Jesus Christ Himself. While some Baptist churches may form an association of like-minded congregations, for the benefit of sharing resources and missionary efforts. But no church governs another church, and there is no overarching parachurch organization that rules over the autonomous local churches.
This paradigm of church conduct is generally not well understood by those whose only experience is with denominational ecclesiastic structure, and thus they form opinions of how Baptists operate, by incorrectly applying the model of their own organizational principles to the Baptists, whose doctrines and methods are individualistic and different.
To fairly figure out why the First Baptist Church of Crystal Springs denied this couple's request, and comment on it, one needs a lot more information than given by the reporter in the news article.