Why didn't Paul Preach the Lord's Prayer?
Because he did not have access to the details in the 4 books of the New Testament; they were not written yet. Although he did not mention it verbatim, Paul made indirect references to the Lord's Prayer. Examples:
2 Corinthians 4:16: ...we are renewed day by day.
1 Corinthians 10:17: …there is one loaf… we all share the one loaf.
In fact many other specific teachings and parables from Jesus were not mentioned by Paul. But this didn't mean those teachings were for the Jews only. Clearly Paul was given the high-level plan through the Holy Spirit. And therefore his main objective was to introduce the gospel of Jesus to the gentiles. The concept of an invisible God with a redemption plan through a son named Jesus who died on the cross, was already very strange and unusual for these people to accept; as they freshly came off their idol worshipping ways. In other words, the high-level explaining of God's plan was in itself a major job for Paul with all the interferences from Jews in the area and other gentiles opposing to Paul's message.
The 4 books of the New-Testament, were not available to Paul. The Bible books of Matthew, Mark, Luke, and John were written and published about 100 years after Paul died. So he did not have any new-testament scripture to read, or reference. But filled with the Holy Spirit, he did the best he could to evangelize these regions, without Bibles. He had to stand up in each church and talk to people about the Gospel of Jesus, then rush to the next church. Here is an example of Paul at the church of Corinth; indicating he had no time to spare: "To this very hour we are hungry and thirsty, we are poorly clothed, we are brutally treated, we are homeless." 1Corinthians 4:11. So reciting the whole new testament to these churches, even if he did know all the words, would have been an impossibility due to shortage of time. Therefore he had to keep it high-level. Paul, although had discussions with the disciples of Jesus about some subjects and debates going on at the time, he never saw the details of these 4 books because they weren't written yet.
Many public references confirm this; that the death of Paul was around 68 AD. The 4 books of the Bible were written after that, and became available to the public around 180 AD. This answers the question of "Why Paul did not mention the Lord's Prayer in his teachings." Then you may ask, "what about those early Christians in the days of Paul? They did not pray the Lord's Prayer daily?" Like everything else, God is consistent in His ways. In the same way He granted righteousness to Abraham for example, God understanding that it was a transitional period situation with limited access to scriptures, He likely granted those people the righteousness of Christ. This is not referenced anywhere in the Bible, but for those who study the Bible regularly and know God, we know His personality and His ways, and see this clearly; John 10:27, "My sheep hear my voice, I know them, and they follow me."
That was to be understood as an interim temporary arrangement for those early Christians. Not to be extended to today's Christians. Today we do have the whole Bible available to us. And the words of Jesus in the 4 Books of the New-Testament are very clear. I personally would look there first as most accurate message, as in, from the deity Jesus Himself. Then complementing info comes from the apostle Paul later.
Many protestants going by Paul's message as the one for us descendants of the gentiles, have missed the mark by a mile. They brush-off discrepancies between Jesus' message and Paul's as something like: "Jesus meant that for the Jews; Paul's message was for the gentiles who became Christians later..."! This is so dangerous on so many levels, disrespectful to God, and causing spiritual-death to so many people! Any church who preaches this, makes it no different than the Pharisees who crucified Jesus despite knowing He was the Messiah. All of Jesus teachings are for all people, not just the Jews. And Paul's teaching is a supplement to Jesus'.
In those days, Paul pressed hard the message of "by faith alone; not by works". This was because he was facing a difficult situation at the time, where the general public feeling was that some works of the flesh had to be done for the forgiveness of sins. Add to that the strong influence from the Jews, who kept poisoning the early church with their requirements. Example, you still have to be circumcised, you still have to refrain from eating certain foods, etc.. Paul kept drilling in the message of "NO, it is by faith alone; you don't need to do anything else in the flesh..." This is why we see this statement a lot "saved by faith alone, not by works..." in Paul's teaching. Then the Protestants got a hold of this, and translated it as: "No, it is by faith alone; you don't need to do anything else; nothing at all." And here we are today, after hundreds of years of producing "do nothing" Christians, people are brainwashed with the wrong message, and many are likely spiritually dead.
People need to be vigilant about this, and stop blindly following their church's instructions of "do-nothing", or Catholics encouraging their members to pray to humans. Examine the facts in the Bible, and research online for yourselves, including this website. And know the difference between the "works of the flesh" and "spiritual requirements". Let's not mix up the two. We're talking about our eternity. Please take the time to examine all the facts and Bible references.