I've been thinking a lot recently about how cultural influence has impacted how the church operates. Consider the Sabbath as an example. At what point did God switch the seventh day of the week for the first? While we do indeed see the early church gathering together on the first day of the week, does this mean that they stopped resting on the seventh day? I can't seem to find any evidence that they did.
Perhaps a more extreme example of cultural influence is how decisions are made in the church. My experience has been that most churches use some form of Robert's Rules and majority rule to make decisions, but as far as I can tell this does not fit the command for unity in the church. Oftentimes majority rule leads to decision making based on coercion, threats, or good-old pork barrel politics rather than decisions made for the sake of glorifying God and unifying His church.
The only remedy that I see comes from Jesus Himself when He says in John 10:27-28 that "27 My sheep hear my voice, and I know them, and they follow me. 28 I give them eternal life, and they will never perish, and no one will snatch them out of my hand." I have completely lost any interest in fighting with anyone over church matters. Whether God uses these other churches for the purpose of His salvation is up to Him. What I do know is that that when we share the true Gospel of Christ then the sheep will hear His voice and follow. Jesus has made this promise.
So to directly answer your question of "How can we convince our fellow believers that this was never the New Testament model?"... I'm not sure that we can or should. I very much appreciate the sentiment behind your question and your desire to change hearts to Christ. When someone tells me that they need a couple hundred grand for a new building then that tells me that they are worshipping the facility and those who gather there more than Jesus. If we are teaching the Gospel without all of the "extras" and their hearts are so inclined to hear it then praise be to God for His mighty work! But until God acts, I think that all we can do is to demonstrate how it should be done through our words and actions.