To place a real world example the influence of the Sheika's mimic the rise of superpowers after World War 2 (1) in which an economical power rise and other whiter after an economical crisis that is tied to social problematic; historically speaking in our world an empire has a duration of 80 years which is also linked to how the world goes into crisis or recessions within a similar time-frame.
Generational speaking, within 3 generations we are destined to repeat mistakes that were on the time of our great-grandfathers but if we go on a more contemporary example right now we are repeating mistakes saw within the generation of the 20 which encompass our grandparents. If we analyze the Sheika's behavior towards their world events, they'd been there within almost every crisis.
We are going to place the Sheika's within every possible scenario with the new converged timeline and we are going to place the structure of the classic feudal Hyrule into a constitutional monarchy as it's the idea I could observe within Breath of the Wild because there were hints on how each dependency or province ruled their own land but ultimate they were influenced by the Royal Family of Hyrule.
The Sheika's power within the kingdom of Hyrule was merely economical, within a context they could be seen as some form of Silicon Valley (2) but also besides being the guardians of the royals they were also the people who influenced the political, mercantile and technological advances of the entire kingdom, we can see they were even more powerful in terms of influence than the Royal Family.
The most crucial aspect on their hegemony were their technological development such as the Gossip Stones (3) which they can decode, transmit information but also self-destruct if the situation is need thus keeping the information safe from the hands of the enemies of the Kingdom, in terms of warfare the Sheikas were ruthless and also could had seize the control of the Kingdom because at least in theory they were more powerful than the Royals but it would lead to a paradox what if you defy the reincarnation of the Goddess you swore to protect? Probably the best answer would be in the creation of the Yiga clan.
Why do I mentioned Silicon Valley before? Because the similarities are there, the two with their interaction on technology influenced politics but also on how the kingdom or the country developed (4) towards politics; politics are usually interlinked with how much you can lobby towards an ideal but also how can you benefit from the ruling class of the country.
Silicon Valley as a PR move they usually present themselves as more of nerdy influencer in which they are usually the first ones that give an idea that everyone should take on their homes or the latest technology, but reality is far macabre because we can look at the scandals of Facebook and Cambridge Analytica (5) in which information became the prime treasure in terms of technological warfare.
Talking about Breath of the Wild, we continued with on how the Sheika's are the protectors of the Goddess, but also through the game experience we are shown as spectators on how the influence technology over the millenia with the creation of the Divine Beasts in which it consolidated them as a race devoted to warfare; a hypotethis I have is that the Royal's didn't placed their power based solely on magic or Triforce but on war games.
In terms of military expansion, natural resources are usually the main objective on war declarations or conquest of territories, geographical outpost and appropriation of information tends also to be the related to reasons of war, because with information, with natural resources and conquest of territories you can become richer, lets think of oil but also precious mineral as our in our world and in between all of that the Sheika's where always there and they are the reason why the Hyrule Royal Family became extremely powerful.
Probably at the end of the converged timeline there wasn't too much information, but there are hints on why the empire fall and that's the lack of coordination to keep the socio-economical power of the kingdom at check