"“Take heed therefore unto yourselves,” the apostle admonished his brethren, “and to all the flock, over the which the Holy Ghost hath made you overseers, to feed the church of God, which He hath purchased with His own blood.” If ministers of the gospel were to bear constantly in mind the fact that they are dealing with the purchase of the blood of Christ, they would have a deeper sense of the importance of their work. They are to take heed to themselves and to their flock. Their own example is to illustrate and enforce their instructions. As teachers of the way of life they should give no occasion for the truth to be evil spoken of. As representatives of Christ they are to maintain the honor of His name. By their devotion, their purity of life, their godly conversation, they are to prove themselves worthy of their high calling. AA 394.1"
"Take heed therefore unto yourselves, and to all the flock, over the which the Holy Ghost hath made you overseers, to feed the church of God, which he hath purchased with his own blood."
The EGW passage directly quotes Acts 20:28 and uses it as a foundation for discussing the duties and responsibilities of ministers in caring for the church.
"Feed the flock of God which is among you, taking the oversight thereof, not by constraint, but willingly; not for filthy lucre, but of a ready mind;"
The EGW passage directly quotes Acts 20:28 (the verse referenced in the AA footnote for this passage, which is more accurate than 1 Peter 5:2) and expands upon its meaning for ministers. The connection is strong and explicit.