Jesus teaches through a parable that the shepherd risks leaving the healthy sheep to find the one. The Pharisees are the healthy sheep.
When Jesus said..."forgive them Father for they know not what they do"...he was not speaking about the Jewish religious leaders, I assure you.
Jesus criticised Tradition -- the belief that the spirit passed from teacher to student. In other words he was against things like apostolic succession and seminaries. He considered them I think to be corrupting influences. He is not against having traditions but against the Tradition discipleship method. This he calls on the carpet and why because he is saying it is time for realization of Jeremiah 31. He is saying that old method of learning about the way is no more, that it is no longer best. He denounces Tradition, so of course the Pharisees seek to embarrass him.
I think he went way further than that. There was no "apostolic succession" or seminaries for Christians in the first century...there was a foretold apostasy however that was already "at work" before the apostles passed off the scene. ( 2 Thessalonians 2:6-7) It grew into a juggernaut of fake Christianity that ended up engulfing the world. It is the "weeds"....just as Jesus said.
So no we do not need to correct each others doctrines, because that is not how things ought to work in a Jeremiah 31 World.
The correction comes from the inside, not the outside. We first have to take in knowledge and then we process that knowledge and either adopt it or reject it.
In the case of Israel, many misunderstand what it means to be a "Jew" in God's eyes.
Paul wrote..."For he is not a Jew who is one on the outside, nor is circumcision something on the outside, on the flesh. 29 But he is a Jew who is one on the inside, and his circumcision is that of the heart by spirit and not by a written code. That person’s praise comes from God, not from people."
John the Baptist also had very stern things to say to the Pharisees....Matthew 3:9....."Do not presume to say to yourselves, ‘We have Abraham as our father.’ For I say to you that God is able to raise up children for Abraham from these stones. 10 The ax is already lying at the root of the trees. Every tree, then, that does not produce fine fruit is to be cut down and thrown into the fire."
What 'fruitage" did Israel produce?
Paul also called the members of Christ's congregation of both Jew and Gentiles, "the Israel of God". (Galatians 6:16)
As he said in Acts 15:14...."Simeon has related how God first concerned Himself about taking from among the Gentiles a people for His name."
Spiritual Jews are not just Jews born of a Jewish mother.....fleshly Israel was cast off and a new nation chosen who would produce the right fruits. Israel as a nation, never did. (Matthew 23:37-39)