Galadriel's intentions.

[Keeper of Dol Guldur]

I wonder then, if Galadriel realized that it would take Boromir's death for Aragorn to resolve to go to Minas Tirith, now that Gandalf had fallen.

Because Aragorn had always intended to return to Minas Tirith with Boromir ... but the second Gandalf was lost and he was forced to become guide to Frodo, he sort of got lost in the task at hand, and obviously made many comments about not knowing which road to take.

I am not Gandalf, and though I have tried to bear his part, I do not know what design or hope he had for his hour, if indeed he had any.

Aragorn then suggested that he and Gimli accompany Frodo and Sam, while the others went to Minas Tirith. But he only suggested this out of loyalty to Frodo.

What good would Anduril and the returning King be in Mordor? Or what use would Gandalf the Grey have been, for that matter? Gandalf couldn't have concealed his power from the malice of Sauron, not once he entered Sauron's lands.