You have to work pretty hard to take the perspective that Vincent dies.
Vincent nearly fails the treadmill scene because he is on the edge of panicking under the stress of being revealed to be prime suspect in a murder case that might put him in jail wrongly and will blow his life's dream of getting to Titan.
It doesn't matter if they check his ID at Titan, whatever that means. He commented he is outside the police's jurisdiction on the way to Titan, he will have achieved his goal regardless of his publicly known validity after docking at Titan.
The line about going home is to illustrate that he is in fact, going home. He was always meant to be among the stars and found regular everyday life difficult. Through his teen years he studied astrophysics, he was an outcast. His home was, is and finally arrived among the stars. Vincent achieved exactly what he set out to do and so did Jerome. Jerome was upset that he was not the best swimmer, as he was designed to be and deliberately chose to walk into traffic. Jerome unable (or had no desire) to escape his own self-resentment, burns in it. Both Anton and Jerome suffer for defining themselves by their biology and the accompanied social expectations.
Jerome passes Vincent his hair as the last expression of his time on Earth. Jerome will no longer grow any hair, a natural measure of time. He has ceased to exist in a physical sense. His mission is complete in giving enough of his valid biological matter for his life and Vincent's life across time.
It is questionable what Vincent's goal in space is to do from a relational perspective.. He saved Anton from drowning twice and as lifted into space on the back of his own effort and everybody that let him pass through the system illegally. His father let him past the final checkpoint, Jerome gave his body, Anton gave him a gap in police authority, Irene did not expose his invalidity and the mission director pushed for the launch timing slot in the face of authoritative danger. What does Vincent give back for all this. A change in spirit from biological determinism to willpower over statistical fact. Not much in comparison to his launch.
Musing aside, Vincent lives.