It is rather doubtful to say that the Greek was thus translated improperly. Those words fit the Greek quite well. Your argument is with interpretation, not translation. And that is why the Word must be subjected to exegesis, not the eisegesis that you accuse the translators of, that when the text is treated to bring out the exact sense of it, the exposition of it will bring unity, not strife, among Spirit-filled brethren. A bishop is indeed an overseer, and usually one who has exercised discernment long and well enough to be an authority in the instruction of fellow bondslaves of Christ.
As far as kings are concerned, in the culture of the Redeemer there is only one King and one Kingdom. A local church is not a democracy, it is an example of a representative republic, where the leaders' tasks are to be prophets carrying God's marching orders to the still-developing seekers of holiness, while at the same time engaging in intercessory activity presenting fellow-believers' needs to The Father and His Son, in both cases as a servant of both God and the assembly of believers, all under The Spirit's guidance..
“Those words fit the Greek quite well.”
Not really. The word translated “church” simply means gathering. It is used in the NT for a riot. No connotation of a top-down church.
Same for “bishop”. It means elder or overseer, which is why modern translations use those words - except the NKJV. They want to echo the KJV.
There is no basis for believing Paul envisioned a medieval “bishop”. King James wanted to translate things “High Church” because, without a hierarchical church, his divine right to rule would come into question.