How would have the investigations and theories about Titanic changed had Jack Phillips (senior wireless officer) survived the sinking?
We would know Titanic's real last transmission, which is one of my personal biggest mysteries about the sinking. I don't think it was "Engine room full up to boilers." I suspect his real last message, even if it was tapped out with the transmitter dead, was something like "goodbye, old man."
Phillips would have had to account for at least two major mistakes on the night of April 14. First, the ice warning from the Mesaba in the late evening, which was not relayed to the bridge. It was not a properly formatted message -- it did not have the MSG [Master's Service Gram] prefix that required the message to be relayed to the captain -- but it was an ice warning nonetheless. Charles Lightoller's memoirs, Titanic and Other Ships, faulted Phillips severely for failing to relay the Mesaba's message. Second, Phillips' message "Shut up, keep out, I am working Cape Race" to Californian, when that ship was announcing it was stopped in ice some 10-20 miles from where Titanic would hit the iceberg.
IOTL, things were relatively friendly between Titanic's surviving officers and Harold Bride. If Phillips had survived, things might not have been so friendly. It's easy to imagine Lightoller and the other officers faulting the wireless crew for not relaying the ice warnings, and Phillips and Bride being questioned about the officers' failure to reduce speed given the ice warnings. I personally don't think the captain and officers took radio seriously; they viewed it something like someone might have viewed email in 1995.