I loved that scene, and it made the scene with him and Aragorn at the end that much more powerful and poignant!
That scene between Boromir and Aragorn is needed.
In Rivendell, we see the Aragorn is afraid that he has inherited Isildur's weakness for power. Arwen tells him that's not true.
Then in the extended scenes, we see that Aragorn is relunctant to be king because he has never been tested to see what strength lies in him. Oh, yes, he's done brave things... but this is a testing of character.
At Lothlorien's parting, Galadriel says the timing for testing, for choosing has come. Aragorn must make up his mind to rise about his fathers failures or to sink lower.
Then comes the new scene between Boromir and Aragorn. Boromir is right. Aragorn is judging all men on past failure. Because Aragorn believes that all men are weak. That's why the ring can't go to Gondor. He sees in Boromir everything he believes about himself...
Until, he passes the test and lets the ring go. There it was... all he had to do was take it. The ring was calling his name. He let Frodo go.
Boromir's death scene is so poignant -- now that all the shadings and nuances have been woven into "my brother, my captain, my king".