That is one aspect that I have found more frustrating about the OT is how Leia gets treated and becomes far more passive or reactionary rather than proactive or a leader, which she really should be. Leia should be in Mon Mothma's role, and possibly more hawkish in her approach to the Empire after Alderaan.
So, I think that some minor adjustments to the story could have Leia be far more proactive and anti-Empire.
That's pretty much where they're going with Leia in the new trilogy, but I don't agree that this should have happened in the OT. For all her fiery Skywalker brashness, she was still raised by Bail Organa, who was a firm believer in democracy, diplomacy and compassion. I don't think she'd want to respond to his demise by betraying the principles he held most dear.
There's actually a few issues in the new comics taking place between Yavin and Hoth where she becomes so blinded by hate for the Empire that she pushes things way too far in an attempt to capture Vader. By the end she sees the cost of revenge isn't worth it.
What exactly was Leia's role in the saga? Was she a military leader? A political leader? Or was she involved with Rebel Alliance intelligence? Lucas seemed to have her involved in all three. And for an large-scale resistance group like the Rebel Alliance, I found that illogical . . . even in a story.
The structure and rankings within the Rebel Alliance isn't made explicit because the war itself is not the focus of the story, the characters are.
We may get some clarification in Rogue One and Rebels, but as of right now my best guess is that in ANH Leia is an agent, courier and mid-level organiser working with Bail. When he's killed I'm thinking she inherits leadership of and/or responsibility for the Alderaan sponsored cells, like the Phoenix cell.
Remember that the Alliance is a collection of various rebel groups from across the galaxy. Some like Saw's are veteran guerilla fighters, others are idealistic kids, some are deserters, dissidents or activists. That's a lot of disparate groups and I think it would be a mistake to go with the old EU's interpretation that the Alliance was one big cohesive military organisation. In some ways it's probably closer in a sense to the way a medieval army was organised with the various units essentially owned and commanded by the nobleman who've sworn an allegiance to a monarch. In theory that puts those men under the King or Queen's command though in practice their loyalty will typically lie with their own lord and if said lord decides to quit the field, his men go with him and bugger what the monarch says.
As such I would image there's not supposed to be any one leader of the Alliance, but a council of representatives, military advisers. If so then Leia would be one among equals, but Mon is the one most (Leia included) look to for leadership. With good reason though since by the time of Endor, Mon Mothma seems to have become the most prominent leader by dint of probably being the last of the
founding Senators still free and/or alive. Also IIRC the first to openly speak out against Palpatine, so she's probably been living in exile the longest.