Baptists are historically descended from the Puritan movement in England and theologically from the continental Anabaptists. Both movements were Protestant.
Not mythical or mysterious at all. I am at least baseline familiar with the unregenerate Patristics -- Ignatius (baptismal regeneration); Cyprian (purgatory, extra nulla salus ecclesiam); Irenaeus (salvation as a reward for works, rather than works as a result of salvation; vs. Eph 2:8+); Origen (more purgatory, a second chance after death for unsaved and devils; vs Rev. 20:10, 15); Augustine (prayers for the dead, university within the visible church), etc., etc. All these are a result of incorporating pantheistic Platonism to fortify doctrines unsound and unproven not only to the NT writers, but also to regenerated literal-grammatical interpreters of this age.
The ante-Nicene "Fathers" stood nearer to the LORD's incomparable never-enlarged, never-augmented, nor amended finalized foundation of The Apostles and prophets (Eph. 2:19-22, 1 Cor. 3:19-11) than we do, but their writings lacked the Go-breathed beauty, clarity, precision, and infallibilty of That Which Is Perfectly Completed, the inscripturated essence Of The Jehovah, The Lord Jesus Christ. Therefore, I would never use their arguments to mask, amend, replace, modify, or buttress any argument that will not pass the naked double-edged Sword of The Spirit, The Hrema of The God, which goes out of His mouth.
Yes, Ignatius did refer to catholicity, but it was not until Constantine married the already prevailing but unscriptural Nicolaitanism--of Ignatius and Cyprian--to the Roman State that we got what is titled the Roman Catholic (made proper) State Church. This raised the hypothetical slippery improvidential invisible catholicity in an impure church, to a very formal embodied visible unity, and that is how Roman Catholicism was birthed to control a newly inaugurated sacral society under the aegis of the civil government.
The Roman Church in America can never rest until She has taken over the operation of the civil government, IMHO, for She is not designed to operate any other way. (BTW, this the point of why Mohler remarked, "...Again, for an evangelical Protestant to respond to the papacy, we have to say that one of the problems is this critical claim to both temporal and spiritual power. And that is a very dangerous mixture. Who knows in what role he is speaking?")
With this the true NT ecclesiology can never find common ground with Roman Catholicism nor Protestantism. This knitting of churchianity with civil government in Statist Catholicism or Protestant Reconstructionism is the very antithesis of the New Testament Church and its Baptistic distinctive; that is, separation of church and state (Jn. 18:36 -- "... My Kingdom is not of this kosmos ..."; Mt. 22:21).
Again, I am not indicating this in a pejorative sense or to open old wounds (as is the wont of many debaters); only to show that my stated conclusion is that you and I will not arrive at an agreeable settlement -- only a restive troubled truce.