There's an interesting comment on the link between those last verses of the prophet Malachi (written some 400 years before Jesus was born) and the words of Jesus you quote (written circa A.D. 50 to 70) in the New International Study Bible 1987 edition. It says regarding Malachi 4:6 the following.
"According to Lk 1:17 John the Baptist sought to accomplish this..."
It makes the link to chapter 3 verse 1 where God promises to send his messenger who will prepare the way before him. Then, suddenly, the Lord being sought by Israel would come as messenger of the covenant. The footnote says,
"This is fulfilled in John the Baptist (see Mt 11:10; Mk 1:2; Lk
If this interpretation is accepted, we can follow through on it to see how there is no contradiction between these old testament and new testament (covenant) texts. They are consistent.
John the Baptist was born before Jesus and prepared the way for Jesus' ministry. John urged the Israelites to repent and to be baptised in readiness for the Messiah. This had a uniting effect in Israel. It created a sense of expectation and hope, so that when Jesus did start his ministry, people were alert to that and responded quickly. Jesus himself said that John the Baptist was the one about whom it was written, 'I will send my messenger ahead of you, who will prepare your way before you.' That's in Matthew 11:2 - 19.
Shortly before then, Jesus sent out the twelve disciples with that warning about families being split due to him. He quoted from Micah 7:6 which showed family disintegration at a time of judgement from God. When Jesus came and quoted that text, he was not present in judgement - "For I did not come to judge the world, but to save it" - John 12:47. However, even while he remained on earth, he caused division because people had to take a stance regarding his astonishing claims. He could not be dismissed indifferently. People were either for him, or against him, and that caused divisions, in families too. Even his own family, at one point, thought he had gone mad and came to try to take him away! See Mark 3:20 - 22.
So, while a positive response to the claims of Christ brings peace to those ones, others who object to his claims will be sharply divided against believers. This can cause division in families. Equally, families who believe in Jesus are united.
As Malachi ended his prophetic writing with warning of a curse, so the last book of the new testament ends with a warning about consequences for not taking God's prophets and prophetic word seriously. We have to decide where we stand on such issues, and those who decide to trust in Jesus often find any unbelieving family members against them.