Something that I don't often see in the "was Edelgard's war necessary" discourse is the interdependence between the Church and the nobility. The question is always framed such as "are Edelgard's reforms worth starting a war over" or "is the net result of the war positive or negative" or "did she need to invade the Kingdom and Alliance to do her thing or was she just a cold-hearted, power-hungry tyrant" and the arguments given for or against either side of the conflict very rarely includes this simple fact :

No reform is possible in Fodlan because the Church and nobility feed off of each other.

It doesn't matter whether or not the specific reforms Edelgard wants to implement deserve thousands of people to die for them (if such a thing is even objectively quantifiable) ; it doesn't matter what the "net result" of the war is (if you can even make an objective measure on that) ; it doesn't matter whether or not having the Kingdom and Alliance under her authority was necessary to make the reforms she wanted ; absolutely no reform, whatsoever, in any territory, could have been achieved and lasted with the Church still standing. Why ?

Because the Church and the nobility enforce each other's "divine right to rule" and neither wants to lose their power.

Rhea wants to keep Fodlan in stasis until she brings Sothis back. She's done that for a thousand years and will keep doing it for as long as she needs to. How did she do it ? By making the powerful (the Crest-bearing descendants of the Ten Elites) dependant on her. She made their strength (their Crest) dependant on properly following and supporting her Church, instilling in Fodlan's societies the belief that a family won't produce Crest-bearing heirs any longer if they oppose the Goddess. The Crests are therefore both a mark of the Goddess' favour, granting nobles their "divine right to rule", and a pair of shackles binding them to the Church. If the Church falls and the commoners don't believe in the Goddess any longer, they also won't believe in the nobles' divine right to rule anymore and will rebel against the caste system. Therefore, the nobles need to enforce belief in the Goddess by showing how devoted they are, as Ferdinand and Lorenz explain is their noble duty to guide the commoners on the path of the faithful, and by giving the Church money to keep its influence and prestige ; otherwise, the nobility itself will be at risk.

But the reverse is, therefore, also true : if the nobility were to fall or stop supporting the Church (the one Rhea leads, mind you), it would lose power, influence and the money necessary to maintain its presence throughout the continent. The Church needs to make sure that the nobility system either stays the same or at least still relies on the belief in the Goddess. Any reform that abolishes the nobility, title inheritance or Crest supremacy is bound to be an absolute nightmare for the Church. Sounds familiar ?

Three Hopes touches onto this by showing us Rhea's attempts at killing the Bishop of the newly restored Southern Church. She doesn't try to get him killed because Edelgard was the one to choose him, she tries to kill him because this Southern Church represents a threat for the Central Church she leads, and she doesn't accept that.

How is the Southern Church a threat ? Two possible, non-exclusive ways :

The Southern Church's teachings are in opposition to the Central Church's teachings : we don't know exactly what those teachings are, but if the Southern Church's teachings imply that the Central Church's ones are fake, it would put their legitimacy into question. Which brings us to the second point :

The Southern Church is trying to become more prominent than the Central one : the Southern Church was dismantled about 200 years before the start of the game for being "dissident" and is now being revived with Edelgard's support. It doesn't even matter what it teaches the people : it fills a gap in the Imperial citizens' lives. By doing so, and by being morally, publicly and financially supported by the new Emperor, it earns the support of the faithful of the Empire and takes it away from the Central Church, in part at least.

So, because Rhea doesn't accept power being taken away from her, she attacks and tries to kill the "rebellion" in the egg, the same way she did with the "dissidents" of the Western Church in Houses, where she sent her students to execute and capture the priests, then executed the survivors without a trial. Sending an assassin against a bishop is clearly not something she would hesitate to do.

But, in this example, the Church was the one "under attack". You might wonder how Rhea would have reacted if Edelgard hadn't restored the Southern Church or started the war and had simply reformed the Empire without bothering anyone else, right ? Well, the answer is : basically the same thing.

What we know of Edelgard's reforms is that they are meant to end the caste system, let anyone reach a position of power, oust the people who are not competent enough to handle them and deny the very idea that birth determines someone's right to rule over a region or a whole nation. Those are all ideas that the Central Church has spent the last millenia denying and fighting against so the nobles would stay dependant on them ; if Edelgard's reforms were to be implemented, and moreso if they were to be successful, therefore proving that Crests and the ability to rule are not related, the Central Church's teachings would be proved wrong and the people of Fodlan, Kingdom and Alliance included, would rebel and stop supporting the Church and nobility. Rhea would lose her power and Fodlan would evolve, one way or another, outside of her control. She would do anything to avoid that, including sending her knights to support the rebellious nobles (like the ones we see in Hopes) who are bound to fight against Edelgard's tyranny to preserve peace and order in Fodlan, even if it means fighting against the Imperial army until they can march on Enbarr and behead the mad tyrant... which is roughly what happens in SS where the Black Eagles form a rebellion and ally with the Church's forces to fight against Edelgard and ultimately defeat her in order to keep the status quo. Yes, they fought this war because they thought Edelgard was in the wrong and needed to be stopped, but the result is the exact same.

(if you want to get into real world parallels, the Catholic Church was in much the same position of power as the Central Church for a good millenia (the xenophobia and the way it wormed itself into being central in every government by saying the King ruled out of God's will and had to be a good Christian if they wanted to keep their throne is especially similar), but then Martin Luther called it out on its corruption and started the Protestant Church (Southern Church, in Fodlan's case), leading to a long religious and physical conflict where the Catholic Church tried to eradicate the dissidents, with fire among other things)

So, what do we deduce from all this ? Three things :

The Church and nobility feed off of each other to maintain their own power ; if one were to disappear or stop working the way it did in the past, the other would collapse.

Rhea is more than ready to do what it takes to preserve her power and influence over Fodlan because it allows her to keep it in stasis and she believes that is what she should do until her mother comes back from the dead.

Edelgard's reforms would put a dent in Rhea's power, war or no war, which would result in Rhea trying to get her killed so she could put a good, obedient noble on the throne in order to restore the Central Church's power and maintain the nobility's standing in the Empire/Fodlan.

So, in conclusion : no significant change to the system can be made and kept in place for more than a few years, a decade at most, without national level of armed conflict at the very least because the Church and nobility will ally and fight tooth and nail to preserve their power. Taking apart the nobility cannot be done without destroying the Central Church at the same time, for one will always try to save/bring back the other to justify its own power. If Edelgard hadn't started the war, Rhea would have ; if Edelgard hadn't invaded and conquered the Kingdom and Alliance, their nobility would have attacked her in order to avoid her reforms giving ideas to their own population, or the population would have rebelled and started a bloody civil war in order to win their freedom, which would have likely caused more casualties, especially civil ones and children, than a fight between two actual armies.

There would have been war anyway. There would have been thousands of deaths anyway. Edelgard did what she did in order to gain the advantage by attacking first, improve her odds and shape the conflict in a way that would cause the least civilian casualties. We can still debate whether or not she was right or wrong, whether or not there was a better way to achieve her goals and save more lives, but the fact remains that while she did start the war, if she had not, someone else would have in order to stop her reforms. There was no peaceful way to change the whole system. As always, when you want to take power away from a group of people, you have to fight for it, and they will fight back.