LRT I really like a lot of the analysis of Y'shtola and Matoya there, and I don't want to append a bunch of speculation/interpretation onto it, but there's just... so much about Matoya as a character that lives in the margins and background of the story that I feel is worth examining.
To start with: Her conflict with the Forum.
This is the very first thing we learn about her, and no one ever goes into any details about it, but there's enough we can kind of piece together from the MSQ to paint a picture. A few of the things she says to you if you visit her cave after the MSQ.
Post-Heavensward: I don't care much for politics. Too much talk and not enough action.
(After: For those we can yet save) Noble causes, hmph. I can't count how many times I've seen what began with the best of intentions be twisted into something altogether different in service to a “noble” cause.
Post-Endwalker: Like old Louisoix, you lot set out to save the world. Like him, you believed in your cause with all your heart. And like him, you succeeded. The spineless wretches of the Forum could stand to take a cue from your example.
We know that she was charged with designing a weapon to use against the Garleans that the Forum decided wasn't worth developing and then opted to flee the continent and abandon everyone there to die. That this--the refusal to act, even in their own defense--is the philosophical difference which keeps her from returning to the motherland seems pretty clear.
I suspect that this is the core of her rivalry and disagreements with Louisoix, Galuf, and Montichaigne, as well. But I'll get to that more later.
The second thing we know about her concerns the Crystal Eye.
How and why she obtained it is never explained, but I am an old-school FF player. I know that she has it because Matoya in the original Final Fantasy game had a crystal eye that was stolen. However, in this case, the eye has a lot of interesting implications.
She tells us that it's an old crystal of light, and therefore we know it is implicitly connected to Hydaelyn. The only person who can use it (aside from Matoya herself, presumably) is Krile. Who has the Echo and the gift of hearing the soul. And then, of course, we soon learn that Matoya is the last keeper of the Antitower, the Sharlayan device for speaking to Hydaelyn. The crystal eye was probably used as part of the Antitower originally.
Matoya, therefore, is an oracle. We can assume she, too, has the Echo like Claudien in Old Sharlayan, and occupied a position like his during the days of the Sharlayan colony. She was the researcher responsible for managing their studies of the Aetherial Sea and for trying to communicate with the Mothercrystal--a thing only those who have the Echo can do.
Which means that, in all likelihood, she knew about Hydaelyn's warning of the Final Days to some degree or other.
This sets her up, in many ways, like Zero. She's effectively meant to be a Warrior of Light, but nobody around her will do anything. She's alone. Watching things fall apart.
Her frustration with the Forum--with her entire culture--was probably not limited to their policy on Garlemald. That was almost certainly just the final straw. Her closest friend/rival seems to have been Louisoix, Sharlayan's foremost expert on prophecy. Followed by Galuf (we know that she kept in touch with him enough to hear about G'raha from him) who dedicated his life to trying to stop the primal Eureka from destroying everyone. It's not hard to imagine that this group of archons, all of them interested in stopping calamities, clashed with the Forum in different ways, but also with one another over how to act on behalf of the star.
When the moment came, and the Forum decided to give up on Etheirys, Louisoix followed the order and went home. Galuf stayed on his island prison. Nobody stood up for the star.
Louisoix thought, perhaps, that he could change things from inside. That seems to have been the root of his many arguments with Fourchenault. He tried for too long to get Sharlayan to take action, and was forced, in the end, to make a desperate last stand. Galuf and Montichaigne seem to have had similar stances. Trying to work within a broken system to achieve change. It didn't work. Galuf follows Louisoix in making a desperate sacrifice to avert calamity. Montichaigne would have gone along with the Exodus.
I think her refusal to return to Old Sharlayan was also her way of saying that she unequivocally rejected the Great Exodus, as well.
She would have been a child at the time, she couldn't have had much say in the matter, but nevertheless, she obediently went back to the motherland to finish her education with the fools who wanted to sit around debating over whether or not it was worth trying to fight for their own lives. No wonder Matoya felt bitter about it.
And just look at how often this is the point over which Y'shtola clashes with others.
When we are busy doing ridiculous fetch quests for the Company of Heroes, she's furious about this. Why do you need a test? Someone needs to do something, doing nothing serves nothing, you idiots! Thancred's indecision about Minfilia, Urianger's waffling and deception over even telling the WoL something is wrong with them, she has no patience for this. Act. Stop being a coward. No amount of thinking about it will make the problem disappear. This is Matoya's influence, through-and-through.
It's ironic, but oddly fitting that Matoya, keeper of the device for communicating with Hydaelyn, struggles to communicate with her apprentice. But so much of her anger and frustration with her friends, colleagues, leaders, and nation stems from their propensity to talk too much and act too little. They heard Hydaelyn's warning and took the exact wrong approach. What good is knowing without doing? What good are words without deeds?
And we see her act. Every single time Y'shtola comes to her for anything, she complies. The plans she locked away forever? Of course you can have them. Keys to the Antitower which nobody should ever use? Go ahead. Her precious older-than-antique Crystal of Light? Fine, borrow it. Something bad happens to Shtola? She will leave her cave for the first time in twenty years and come take care of her. Oh, you're making pigs fly? I'll open up the evil magic lab.
She is the most ridiculously doting master in the entire game, she just tries to cover it up (badly) with crochety remarks.
