I think you would be wrong that those Baptists believed "every believer is the church." They had a very high view of the church, often undergoing persecution for illegally assembling. They defended autonomy of the local church to the hilt, maintaining that God had endowed each congregation with all the gifts it needed to fulfill its mission. ("To each of these churches therefore gathered, according to his mind declared in his word, he has given all that power and authority, which is in any way needful for their carrying on that order in worship and discipline, which he has instituted for them to observe; with commands and rules for the due and right exerting, and executing of that power." Second London Confession)
Their beliefs sprang more from a belief in the priesthood of the believer and a distrust of sacerdotalism.
Those early Baptists also did not come down quite clearly on whether baptism and the Lord's Supper are ordinances of the church or of the New Testament. (The Second London Confesses edges toward the former on baptism, I admit.)
While you seem to draw very fine distinctions in the terminology used for the church, the early Baptists often were not quite that specific. And yes, the signers of the Second London Confession explicitly upheld the concept of the universal church. The First London Confession does not explicitly endorse the concept, but it holds that "And although the particular congregations be distinct, and several bodies, every one as a compact and knit city within itself; yet are they all to walk by one rule of truth; so also they (by all means convenient) are to have the counsel and help one of another, if necessity require it, as members of one body, in the common faith, under Christ their head."
You are certainly welcome to air your opinions, but this is, after all, a history forum, and I approach questions from that angle.