While the phoenix and grail can be viewed as symbolic or representative of the Savior's power, we have His actual power here on earth - it's not mythology. The Lord gave to his apostles power to act in His name and to do the works He would do. In the book of Acts we read stories of the apostles healing and otherwise continuing the miracles that the Savior did (e.g, Acts 3). This priesthood power was taken from the earth when the apostles were killed. Different branches of the church quickly apostatized. The priesthood was not found on the earth (other than the three Nephites and John the Beloved, all who remained to watch the earth and to prepare for the restoration of the Church) until May 15, 1829 when John the Baptist conferred the Aaronic Priesthood upon Joseph Smith and Oliver Cowdery. Shortly after this the Melchizedek Priesthood was restored. Once again, the Lord's power and authority was upon the earth. Once again were men able to act in the name of the Lord and perform miracles in His name. The priesthood is far greater than the power of the phoenix or the Holy Grail (ignoring the fact that they are mythical). Those who have been given the priesthood hold real power and authority.
Those who have the priesthood have a portion of God's power. They should strive to be like Him in faith, in righteousness, and in holiness. One of the most important and sacred names for our Father is Man of Holiness. When we are commanded to "arise from the dust and be men" we are commanded to be like the ultimate Man (not intending any disrespect) - our Heavenly Father. We should strive to live as our Heavenly Father wants us to live. As we shake ourselves free of the dust of mortality, as we strive to remain free from the dirt and the mud of sin, we can better prepare to live with our Father again. We can, like the phoenix, arise from the ashes and dust of our past to a new life of purity.