Was Jesus in unity with the Holy Spirit?

The Apostle John tells us,

[Jesus] came to His own, and those who were His own did not receive Him. But as many as received Him, to them He gave the right to become children of God, even to those who believe in His name (John 1:11-12 NASB Updated).

These words summarize quite well the evolution, if you will, of Jesus' ministry to a world of lost people. Jesus came first to the "lost sheep of the house of Israel" (Matthew 10:6, the verse which follows the one you quoted), since he himself was of the house of Israel through David's kingly line (Matthew 1:6 ff.).

In other words, the Jewish people with whom Jesus identified quite naturally were the first people-group to whom his kingdom was offered when he appeared on the scene in first century Palestine. After all, Yahweh's promise of a Messiah was given to Israel through Jewish prophets and Jewish Scriptures.

By and large, however, the house of Israel rejected Jesus as their Messiah, as John observed when he said "His own [people] did not receive Him." Eventually, then, particularly with the ministry of the apostle Philip and later the apostle Paul, the church which Jesus planted on the Day of Pentecost became open to all comers, whether Jew or Gentile, slave or free, male or female (loosely, the substance of the verse you quoted--Colossians 3:11).

We must note, however, that Jesus did not limit his reach to the lost sheep of Israel, though they comprised his primary audience. We need only observe how on occasion Jesus went out of his way to offer his salvation to those outside of Israel's fold. The Samaritan woman at the well was one such person, and the seed which Jesus planted in Samaria through her bore fruit both then and years later, when the apostle Philip was privileged to "preach the good news about the kingdom of God and the name of Jesus Christ" to many Samaritans who believed Philip's message and were baptized and later received the gift of the Holy Spirit through the ministry of the apostles Peter and John.

Notice, too, the way Jesus interacted with the Syrophoenician woman in Matthew 15. When she cried out to Jesus, whom she called “Son of David,” Jesus initially told her in the hearing of his disciples (who wanted Jesus to shoo her away),

I was sent only to the lost sheep of the house of Israel (v.24).

Refusing to give up, however, the woman in her desperation to get help from Jesus for her demon-possessed daughter continued to beg Jesus for help. Even after Jesus told her,

It is not good to take the children’s bread and throw it to the dogs (v.26),

the woman acknowledged her status as a “Gentile dog,” whereupon Jesus commended her, saying,

”O woman, your faith is great; it shall be done for you as you wish.” And her daughter was healed at once.

I could multiply examples of how Jesus went beyond the pale of Judaism with his healing message (e.g., the Roman centurion of whom Jesus said, "Truly I say to you, I have not found such great faith with anyone in Israel”—Matthew 8:10). Remember as well, the words of Jesus which seemed to anger his fellow Jews from Nazareth most were the following:

”But I say to you in truth, there were many widows in Israel in the days of Elijah, when the sky was shut up for three years and six month, when a great famine came over all the land; and yet Elijah was sent to none of them, but only to Zarephath, in the land of Sidon, to a woman who was a widow. And there were many lepers in Israel in the time of Elisha the prophet; and none of them was cleansed, but only Naaman the Syrian” (Luke 4:25-27). [The following verses detail how Jesus' fellow Nazarenes were so enraged with these words that they drove him out of the city and would have thrown him over a cliff had he not "passed through their midst"!]

Even in the Old Testament, God did not ignore outsiders to Judaism, and he made provision for people who were variously called sojourners, strangers, and aliens to join themselves to the life of Israel, provided they respected and obeyed the Law of Moses. Moreover, writ large in the history of God's dealings with and design for monotheistic Israel was for her to be a spiritual light in the midst of great spiritual darkness, with the expectation that at least some people from the nations surrounding her would be attracted to and identify with her worship of the one true God, whose name was YHWH (or Yahweh).

I hope by now my answer to your question is clear! Jesus’ primary audience was the Jews, but at the same time he was reaching out to his own people, he also reached out to those who were not his people according to the flesh. Furthermore, after Jesus’ resurrection and ascension, when the Holy Spirit descended on the Day of Pentecost, the strictly “Jewish church” slowly became the church for all comers, a truth Peter eventually came to realize, saying,

”Therefore if God gave to . . . [the Gentiles] the same gift [of the Holy Spirit] to us also after believing in the Lord Jesus Christ, who was I that I could stand in God’s way?” (Acts 11:17).

In conclusion, while Paul did say in Romans 10:12 (which you quoted) that there is no difference between Jew and Gentile in the Church Universal, also known as the body of Christ, Paul also said in the first chapter of his letter to the Romans,

For I am not ashamed of the gospel, for it is the power of God for salvation to everyone who believes, to the Jew first and also to the Greek.

Jesus said to a Samaritan women:

Our fathers worshipped in this mountain; and ye say, that in Jerusalem is the place where men ought to worship. Jesus saith unto her, Woman, believe me, the hour cometh, when ye shall neither in this mountain, nor yet at Jerusalem, worship the Father. Ye worship ye know not what: we know what we worship: for salvation is of the Jews. But the hour cometh, and now is, when the true worshippers shall worship the Father in spirit and in truth: for the Father seeketh such to worship him.
-- John 4:20-23 (KJV)

"salvation is of the Jews. But, ...", is a clear indication that what began with the Jews would not stop there. Jesus' statement of the great commission at his last meeting with the Apostles confirms that the scope of his vision was not confined to the Jews only, but would extend to the ends of the earth. (Acts 1:4-8).

Paul says:

For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek.
-- Romans 1:16 (KJV)

Paul repeats the expression "the Jew first, and also to the Greek" again, twice in Romans 2:9,10. He also gives an elaborate illustration of it in his olive tree analogy in Romans 11.

Conclusion

It seems pretty clear what Jesus' game plan was: first, give the natural olive branches an opportunity to bear fruit (dig & dung Luke 13:6-9; vine & branches John 15:1-8); second, prune out the branches that continue not to produce fruit, and graft in the wild olive branches.

Jesus is the husbandman, and the Apostles, including Paul, are his workers. Jesus assigned the Apostles the task of trying to get the natural olive branches to bear their fruit. Having limited success with the natural branches, however, Jesus then assigned Paul the task of grafting in the wild olive branches.