The detail that defines the phoenix in modern usage, immolation and rebirth from the ashes, is a later accretion, and it can be watched arriving.
Roman-era writers add pieces. Some describe the old bird building a nest of aromatic woods; some have it die there and a young bird emerge from the body or from a worm generated in the decaying flesh, which is a version much less appealing than the fiery one and was current for a long time. Tacitus discusses the bird's periodicity with visible scepticism about the reported intervals, which vary wildly between sources, five hundred years, five hundred and forty, a thousand, and the Egyptian Sothic period of 1,461 years all appear.
The decisive move is Christian. Clement of Rome, writing to the Corinthians at the end of the first century, cites the phoenix as evidence that God can raise the dead, and the bird becomes a standard proof-text for the resurrection. Once it is doing that work, the version in which it is consumed and returns whole is obviously the useful one, and it is that version the Physiologus transmits to medieval Europe, complete with the explicit allegory of Christ dying and rising. Christian art puts phoenixes on tombs and in mosaics for exactly this reason.
Two cautions follow. The first is that the phoenix should not be equated with the Chinese fenghuang, which does not burn, does not die and is not reborn; the identification is a translator's convenience. The second is that the modern phrase about rising from the ashes describes the latest form of a bird whose earliest reporter explicitly refused to vouch for a much duller story.