"Christ's people should, as occasion requires, organize such associations and conventions as may best secure cooperation for the great objects of the Kingdom of God. Such organizations have no authority over one another or over the churches. They are voluntary and advisory bodies designed to elicit, combine, and direct the energies of our people in the most effective manner. Members of New Testament churches should cooperate with one another in carrying forward the missionary, educational, and benevolent ministries for the extension of Christ's Kingdom. Christian unity in the New Testament sense is spiritual harmony and voluntary cooperation for common ends by various groups of Christ's people. Cooperation is desirable between the various Christian denominations, when the end to be attained is itself justified, and when such cooperation involves no violation of conscience or compromise of loyalty to Christ and His Word as revealed in the New Testament."
In this section of the BF&M we find an articulation of the approval of "associations" and "cooperations" between churches and different Christian organizations, within an overall framework of what is historically a very central doctrine for Baptists: the autonomy of the local church ("Such organizations have no authority over one another or over the churches"; "Christian unity in the New Testament sense is spiritual harmony and voluntary cooperation..."; etc.).
I don't intend to give a full critique of congregationalist church government here, nor to give a full articulation and defense of Presbyterian polity. I'll just say that I believe the ideas of cooperation, co-belligerence, and informal associations are operative even within Presbyterian circles; 'NAPARC', or the 'National Association of Presbyterian and Reformed Churches' is one such association, within which distinct denominations which have no ecclesial authority over one another's churches have voluntarily chosen to associate because of the large overlap of our doctrinal and ecclesial commitments. Moreover, it is not uncommon for many 'NAPARC' churches to engage with, support, host conferences with, and associate in various ways with churches and Christian organizations beyond our official ecclesial bounds, for the extension of the kingdom in various areas. This is wise, so long as it follows the principle articulated at the end of the BF&M section here: "when such cooperation involves no violation of conscience or compromise of loyalty to Christ..."
Commenting briefly on forms of church government, there have historically been three broad approaches, with nuanced different expressions within each: episcopal (rule by singular regional bishops in a hierarchical fashion), presbyterian (rule by groups of elders in graded courts), and congregational (autonomous local churches, led more or less authoritatively by a group of elders usually). Arguments in favor of congregational polity include appeals to the church discipline procedure in Matthew 18 and a strong emphasis on the priesthood of all believers; while the crucial role of the office of elder is usually acknowledged by Baptists, there is a very strong desire to protect the biblical idea that all believers have an "equal footing" in Christ, and that a local, visible body of believers should be sufficient to judge and govern itself in matters of doctrine and discipline. There are probably a lot of other strong arguments for congregationalism that I'm no expert on.
Again, ecclesiology is not my particular expertise, and I would likely lose a debate in this area if arguing with a well-educated Baptist or Anglican, but I'd be remiss if I didn't at least offer a very brief sketch of some overlapping key arguments in favor of presbyterian polity (which can each be debated in much greater detail, no doubt):
a) connectionalism: the way the New Testament refers to visible church bodies includes language that assumes the category of a discernible regional church beyond individual "local" churches; Paul also assumes that individual churches have formal responsibility for the good of other churches, by means of financial and prayer support (this argument supports episcopalianism equally with presbyterianism); Acts 15 also, while not a strictly "prescriptive" passage, shows elders coming together to make authoritative top-down decisions for regional churches (always based on application of Scripture, not arbitrary, rogue decrees--ecclesial authority is only and ever "ministerial" and "declarative" according to the PCA Book of Church Order)
b) terminology: "bishop/overseer," "elder/presbyter," and "pastor/shepherd" are used interchangeably in the New Testament to refer to the same office that carries out the fundamental, authoritative spiritual leadership within in a church body (this may count against episcopalian polity, barring certain interpretations of a few passages of Acts and Revelation, and it has to be admitted that regional singular "bishops" existed very early on in the Christian church era)
c) history of office of elder: groups of elders governing at levels of both individual local congregations and larger regional churches, in a system of graded courts, seems to be the assumed norm of the New Testament (note it was the "apostles and elders" debating the issues at the Jerusalem council in Acts 15 and then handing down joint recommendations, rather than any apostle simply pontificating authoritatively, as would have been an apostle's right); moreover, the office of elder was not a New Testament invention but was instituted first under Moses (with no explicit change in the NT), and involved popular election (bottom-up selection for representation), ordination (top-down installation), and smaller and larger groups of responsibility (graded courts)--see the review in Deut. 1:9-17
d) visible unity: Jesus prayed for Christian unity in the "high priestly prayer" of John 17, and far from it being a merely "spiritual," it is to be an increasingly visible unity, by which the world will be convinced that Jesus really came from the Father; obviously, Christians have a long way to go in achieving this, but it will happen in history (Eph. 4:11-13)
e) one and many: as a more general consideration, it is to be noted that presbyterianism at least sees itself as attempting to hold an ideal balance between the "one" and the "many," because of a) the biblical pattern of God's covenant dealings in Scripture embracing groups of people (families, nations, kingdoms, etc.--even covenants made in one sense with individuals always implicate others as well...consider the so-called "Abrahamic" and "Davidic" covenants which involve Abraham's entire household and David's descendants), in covenant solidarity; and ultimately b) in reflection of the Triune nature of God Himself, who, as one Being in three Persons, covenanted eternally within Himself to carry out creation and redemption, for the display of His glory.
To Presbyterians (and likely to episcopalians of whatever stripe), congregational church polity smacks more of modernist individualism than of a biblical mindset of corporate covenantal accountability. But to be fair, to Baptists, presbyterianism and episcopalianism likely often smack of dangerous authoritarianism and sacerdotalism, compromising the priesthood-of-believers doctrine.
To be honest, there are pragmatic advantages and disadvantages to each system, and every denomination executes the designs of the system they hold to with less than ideal success (or consistency). Baptists organizations sometimes functionally exercise authority over individual churches that formally they don't claim to have, and believe you me, there are Presbyterian churches that find interesting ways around the supposedly authoritative determinations of their presybteries, synods, and national assemblies. But how well a particular system of church government is implemented is not the final criterion of whether it's a good system. Nor is church government to be regarded as "adiaphora" to be determined by mere pragmatic concerns and individual preferences. Some rather specific principles of church government are revealed in Scripture in a prescriptive enough manner that we must study them, argue them out charitably between each other, submit to them, and implement them by the wisdom given by the Holy Spirit more and more, under the one and only Head of the church, Christ Himself.
