"Independent congregations" can go just as apostate just as fast as large organizations. Maybe faster, since there are no checks and balances to slow them down.
They usually grow up around a charismatic individual as pastor. If the charismatic individual is Godly and orthodox, great. If he's not, you get Joseph Smith.
The only good news is that you can leave and and it's easy to start a new "independent congregation".
That's a practical note. As a Biblical note, nothing about the New Testament churches was "independent". That's why Paul and the other apostles could write letters and issue them commands.
Certainly, and they did just that. And, Peter noted that many of the teachers were wolves in sheeps clothing. Clouds without rain.
That doesn't justify a gigantus, wrong-headed organization which intentionally teaches that it eclipses the written word of God, adds spurious doctrines from hell, and leads people off into the darkness. I'll take the "easy to start" congregations anytime. And, the only way to know if a teacher is "orthodox" is to compare him to something, namely the Scriptures. Hmmm.
As to the independence of the congregations, Paul is noting just that when he uses the plural term "also ALL of the assemblies of the Gentiles." in the letter to the Romans (16:4). Notice, he recognizes they are all over the world, not centered in Rome. Rome's egomaniacal stranglehold on its parishoners needs another Reformation from within.