It is impossible to reverse rayshift an entity, so she could not be physically retrieved from the Accel Order singularity. Since actual physical time travel is mostly "impossible", the way rayshifting works is that the physical body remains in the present inside the "coffin" and the "soul-form" of the body is sent into the singularity, where Chaldea and staff work overtime to "prove" its actual existence, fooling the world to think it belongs there. Iri could not have been physically retrieved since there is no way to transfer a physical body from a singularity back to Chaldea (ie. the future).

However as a homunculus and a Grail terminal herself, Iri isn't exactly human, which means there is the potential for Da Vinci to "cheat the system", most likely by creating a new body in Chaldea with available resources. Since she isn't actually human herself, spiritual transferrence in a "reverse rayshift" style into a compatible vessel was probably possible, and she most likely became a servant that way.