For public prayer in modern churches you really need to go back to the Old Testament.
Isaiah 56:7 KJV Even them will I bring to my holy mountain, and make them joyful in my house of prayer: their burnt offerings and their sacrifices shall be accepted upon mine altar; for mine house shall be called an house of prayer for all people.
Technically I cannot think of a publicly led prayer in the Temple. Solomon led a prayer outside of the Temple and Daniel prayed three times a day towards Jerusalem. In Matthew 6:5, Jesus talks of those praying in synagogues and streets, but overall his comment is not favorable.
There are many verses in the New Testament referring to prayer. The first part of James 5:16 is often overlooked.
James 5:16 KJV Confess your faults one to another, and pray one for another, that ye may be healed. The effectual fervent prayer of a righteous man availeth much.
I firmly believe the men who get up and pray should pray Matthew 6:9-13. I cannot think of a time in any church where I have heard this prayed, but it is how our Savior, Jesus Christ, instructed us to pray. When listening to another pray in their own words I have to pause and consider if I agree. There have been times when I just pray Matthew 6:9-13 silently during public prayer.