Who exactly was running the Rebel Alliance? We saw Generals a-plenty, and Leia Organa was high up in their ranks… but who was actually in charge? If they managed to triumph over the Empire, who would be in charge then? If there was a squabble in the upper ranks (and there often was), who was the deciding voice, the level head, the calm voice of reason that everyone could depend on?
That voice, that calm, that leadership came from a former senator of Chandrila. Though only seen briefly in the Skywalker Saga, Mon Mothma could be the most important character in Star Wars.
The animated series, books, deleted scenes, and Rogue One: A Star Wars Story back this up, but let's pretend that they don't for a moment. Going solely by the films in the Skywalker Saga? Mon Mothma is crucial because for two and a quarter movies (in the original trilogy) we wondered who the kriff was leading this whole dance, and following the Jabba escapades in Star Wars: Return of the Jedi, we finally got an answer. The Rebel fleet mustered near Sullust, and the entire leadership gathered. A dangerous thing for them to do, but those were dangerous times.
We'd already met the Emperor in this movie, the wrinkled, cackling, shriveled up Sith-bag who called the shots for the Empire. When Mon Mothma (Caroline Blakiston) stepped forward and began to speak to the assembled Alliance leadership, she proved to be everything that he wasn't. She wore flowing white, she was serious, she knew the stakes, and she spoke with wisdom instead of fear. She instilled hope in us before we even realized it, and her stoic leadership (broken only a little when she mentioned how many Bothans perished so they could have this chance at a surprise attack) gave hope. Mon Mothma was the anti-Emperor, and she was exactly what the Rebellion needed.
Sure, most of the plan that she and the other Rebels concocted was exactly what the Emperor wanted them to do... but shifty Sheev missed a few steps, ignored the living Force (Ewoks ahoy), and the plan — despite it being a trap — succeeded. Without someone at the head of the Alliance deciding that "the time for their attack" had come, then we'd likely have had another group of people arguing about the hows, whys, whats, etc. Not here. Mon Mothma, for better or worse, put the rest of this film in action... and Sheev Palpatine got tossed down a hole. Did she do the tossing herself? Of course not. Mon Mothma knew how to delegate.
This might seem like a flimsy case, so let's bring in all of the ancillary material featuring Mon Mothma that makes it more solid. When the alliance was barely anything more than a ragtag group of militants, she made them into a movement. Secretly working on the political side of things with Bail and Breha Organa (as we read in Leia: Princess of Alderaan by Claudia Gray), Mon Mothma was there from the start. She knew there were problems long before Palpatine seized power, so she had been prepared — deleted scenes from Star Wars: Episode III – Revenge of the Sith include her, Padme Amidala, and a group of loyalist senators (Bail Organa included) preparing an opposition to Chancellor Palpatine's actions. The younger Mon Mothma is played by Genevieve O'Reilly in these scenes, and though her scenes were cut, O'Reilly would don those white robes in live-action again.
Mon Mothma acts as a political mentor to Padme in the book Queen's Shadow (by E.K. Johnson), and supports her throughout Star Wars: The Clone Wars, in which she is voiced by Kath Soucie. She also supports Phoenix Squadron in Star Wars Rebels (this time voiced by O'Reilly), and when the time came to step out from the shadows and make a stand? O'Reilly plays the moment to perfection, going on to step back into live action for Rogue One. She proves to be one of the only people who believes in Jyn Erso — without Mon Mothma, the Alliance would probably not have trusted Jyn. Without Jyn, there are no secret plans, and without the plans, the alliance go boom boom via Death Star in the first film. Ya burnt.
Thankfully, Mon Mothma saw hope... and managed to cut through the bureaucracy to send Jyn on her mission. She's also one of the only members of the alliance to know about Bail Organa's secret Jedi friend.
The real reason that Mon Mothma is so pivotal, however, comes in books that take place after the events of Jedi, when the New Republic is being formed. The events of the sequel trilogy (and the entire rise of the First Order) would not have happened without her actions in books like the Aftermath trilogy (Chuck Wendig) and Bloodline (Claudia Gray). Either through success or failure (mostly the latter), she set it all in motion.
There was no question that Mon Mothma was the one who would lead the New Republic following the Battle of Endor. She accepts that power, but she also takes steps to ensure that one person never has as much power as the Emperor did. She puts forward a strong agenda for demilitarization once the Empire is finished for good, and this goes forward.
The New Republic, as we know, didn't turn out to be all that great. Bloodline shows that it fell astray in very similar ways to the Old Republic, and if Mon Mothma had still been around to lead, things may have been different. The problem is that there was only one Mon Mothma — she was irreplaceable, and her health waned. She wasn't going to be around forever. Everyone naturally looked to Leia to take her place... but then the truth of Leia's parentage came out, and Leia's political career was over.
Whether she was right or not, the cycle of war (and more movies) continued because of things that were set in motion here by Mon Mothma. More importantly, though, is the fact that she acted as a mentor to Leia Organa for most of her life. Before politics and before the Rebellion, she was there. When her adoptive parents perished, she was there. When the war was 'won' and political chaos ensued once again? Leia always depended on Mon Mothma, and she never wanted to take her place.
She ended up having to do so anyway, albeit in a way that she probably didn't expect. Mon Mothma's influence over galactic events, the Rebel Alliance, the New Republic, and Leia Organa herself (Mon Mothma was certainly an inspiration for the Resistance), makes her an extremely pivotal player in the Star Wars galaxy. Remove her from the stories, and all three trilogies would be very different.
And yes, she showed emotion over some dead Bothans. No one else did, and no one else seemed to care. Compassion counts.
Without that? You might as well install another Palpatine.