[the phantom]

is Boromir in the Fellowship for the wrong reasons?

No.
(not unless you think Aragorn also went along for the wrong reason)

All the other members joined to help Frodo out

Not Aragorn. He was also headed for Minas Tirith. But you obviously know that because you said-

I left Aragorn out for this reason. He too originally joined the Company to head for Minas Tirith

Huh? Why is that a reason to leave Aragorn out when it is precisely what you are criticizing Boromir for? He had the same plan as Boromir and somehow his plan is fine but Boromir's is not?

But in Moria the burden of Gandalf had been laid on him; and he knew that he could not now forsake the Ring, if Frodo refused in the end to go with Boromir.

This does not change anything. Aragorn's original plan was to go to Minas Tirith. The burden of Gandalf had been laid on him, not Boromir, so why should Boromir's plans change? Just think- if Gandalf had come through Moria then Aragorn would have been willing to abandon Frodo and go to Minas Tirith (just like Boromir). Would that have made Aragorn's reasons for being in the Fellowship wrong?

No.

If the eastern side was chosen Boromir would presumably have left the fellowship, to the benefit of Gondor (at least during the War of the Ring), and with obvious major changes in the story plot. If the western side of the Anduin one would presume the existent plot would basically have remained intact. I am not at all convinced that it was a wise decision to take to the boats.

Yes, Boro probably would've split had they chosen the eastern bank- but who else might've split?

Remember, when they reached Rauros Aragorn was leaning heavily towards dividing the Fellowship. He suggested that Frodo, Sam, and Gimli would continue on towards Mordor and the rest would go with Boromir (Aragorn planned on staying with Frodo).

So if they had to make that decision earlier who is to say that Legolas, Merry, and Pippin would not have gone with Boromir? That definitely would've changed things.

But if they would've chosen the western bank... how do they get to Mordor???

There isn't a place to cross the Anduin until Gondor so the Fellowship would stay together and likely end up in Minas Tirith. That could change the story in many ways.

I think the boats were a super super gift. It moved the point of no return back a little.