Why was Boromir allowed to join the Fellowship?

[Nogrod]

Coming back to the original question I think you could also see it from the POV of the prof and what he was intending (aka. not only discuss it inside the logic of the story and the psychology of the characters).

It's easy to see that including Boromir is a smart move as it brings interesting tensions, conflicting loyalties, social and personal struggles etc. to the fellowship (and of course opening up the POV of the men of Minas Tirith to the whole mess) - great dramatic possibilities open up to the storyteller by adding a character like Boromir. And the prof really used those possibilities to the max.

But possibly even more importantly, I think, adding Boromir (and what he does) adds also to the general idea, or worldview, that things happen by a sort of providence where every act has it's role. Frodo would not have gone alone hadn't Boromir tried to take the ring - and with Aragorn, Gimli and maybe others with him the trip to Mordor would have been different and the Ring probably would not have been destroyed. Like with Gollum being still alive in the end securing the Ring gets destroyed as Frodo couldn't have done it - or Gandalf falling in Moria without which the two aforementioned things probably wouldn't have happened, Merry & Pip being taken captives and ending up in Fangorn pushing the ents into war with Saruman and thus enabling Rohan to join the fight at the Pelennor fields etc.

So even "bad" things serve a purpose.