So basically the stars aligned and Sylvanas happened to save the horde at the Broken Shore which just happened to give Vol'in a false reason to make her Warchief.
I think you're wrong.
Saying that "the stars aligned" is a bit of an over-exaggeration as the Loas did encourage Vol'Jin to grant Sylvanas the title, as someone who is heavily connected to the voodoo there's no one whos opinion Vol'Jin respects as much as the Loas. Had he lived I think Vol'Jin would still have given Sylvanas the lead although he would have been more hesitant to do so. He still disliked and distrusted Sylvanas even as he gave her the title, I doubt he only listened to the visions of the Loa since he was high or something along those lines from the fell.
edit: additionally, why not tell them of the real reason?
He was on his death bed fighting the fel, he was in a race against time, he died in the middle of his speech and he left us with many unanswered question whether my theory is wrong or correct, he never specified what the Loas said, why Sylvanas and not Varok, Thrall or anyone else for example and that remains unanswered whatever the truth is. He clearly died before he could finish his speech.
Alleria may not like the horde, but she wont let Azeroth get destroyed. Thus she will help them.
Anyone who has read Wolfheart would say the same of Genn Greymane, yet when the Horde retreated he assumed the worst and thought that Sylvanas left them to die just for giggles. Jaina on top of that was the biggest supporter of peace prior to mists of pandaria. Just one event changed Jainas ability to see reason completely, you find it unlikely that Alleria, after fighting orcs for decades, loosing countless friends why would she trust the orcs.
Also, at one point Vol'jin says something among the lines: "Don't let the horde fall" or something. So his concern seem to be for the horde rather than inviting elfies to the party.
This is their entire speach:
Vol'Jin: Windrunner, come forward.
Sylvanas: Warchief.
Vol'Jin: The Loa spirits say death will claim me soon.
Sylvanas: In the end death claims us all, but the Horde will live on.
Vol'Jin: I have never trusted you, nor would I have ever imagined that in our darkest time, that you would be the one to save us. The spirits have granted me clarity, a vision, they whisper a name, many will not understand, but you must step out of the shadows and lead. You must be War... wchief...
Nowhere in that speech is anything remotely resembling "Don't let the horde fall" mentioned.
Lastly, there is nothing preventing Alleria from getting convinced by Sylvanas even if she 'merely' represents the Forsaken.
Ah, but there is, how many times has Sylvanas been invited to join the Horde affairs, once, during the Siege of Orgrimmar, and that was only because they needed her, and even then the role she was allowed to play was very minor. She is being kept out of all Horde business because its not only the Alliance that dislikes her, she is only an ally out of necessity. She needs to be in the center, otherwise the people in the center will keep her out.
Surely, the Warchief title means nothing to Alleria but the feelings for her sister does.
That's true, but if the only sources on Sylvanas would be other people than Sylvanas, such as Vereesa then she would get a very unreal depiction of who her sister was.
The other leaders merely need to give Sylvanas freedom in that one mission, making her Warchief would not be needed.
And why would they give her that autonomy? They have never given it to her earlier, and before you say "the threat will unite all" lets remember that it is the bigotry between the Horde and the Alliance that creates the Order Halls, why would the Horde throw away their bigotry towards Sylvanas if they wont throw it away for the Alliance?
EDIT: Kudos by the way, your post has definitely shaken the legitimacy of my theory, but I'm not quite convinced just yet that it is less reasonable than yours.