This is getting into more of a theological discussion, but I do think it's kinda sad if you can't fathom even an entirely fictional concept of a caring god. Of course, just because a god cares, doesn't mean they can save us.
This is something that's come up around Judeo-Christian theoology, so I'll borrow a bit from their angle. In the real world, it's generally agreed that if an ultimately good and caring god exists, they cannot be all of 'omnipotent', 'omniscient' and 'omnipresent', because we live in a world where bad things happen. So either said god doesn't have the power to stop those bad things (ergo not omnipotent), doesn't know how to (ergo not omniscient), or doesn't know it's going on (ergo not omnipresent), or possibly some combination of these things. I've also found this is an approach that holds true for understanding extremely strong heroic characters, too, and where they may fail; for example, Superman is effectively omnipotent on the scale of 'stopping Metropolis crime', but can't be everywhere at once and may not know how to solve every problem with his skillset; The Flash can be anywhere he needs to be, but might not know what to do or have the capacity to win when he gets there; Batman knows anything that's ever relevant, but is not necessarily capable of getting to the problem or solving it himself.
So... let's look to Hydaelyn. Fundamentally a caring god on the scale the story is focused on (sure, she hates the Ascians right now, but so should/do we, they're trying to blow up our home and don't really care about who that hurts), so why are we in such strife, which of the omni-s is she not? Well, we can safely say she's omnipresent; she's always aware of the situation. But omnipotent, definitely not; we know for a fact she's actually fairly weak on the grand scale. And omniscient, I'd argue not; while we've never really seen limits to her intelligence, we've also never really pushed it, but she's never exactly concocted intricate plans.
Hydaelyn can be a caring and loving god; there is nothing saying that she is not. But just because she cares, doesn't mean she's always capable of acting on that.
Also, I don't think all those other games came at it from a deep theological perspective, so much as a purely practical and cathartic one: in a medium we largely interact with through violence, any big fancy figure you introduce will inevitably be followed by someone wanting to punch that figure out. So most of them go 'sure, why not' and cue you up to go fight any god they introduce, because a lot of people struggle to settle for less than that. Which I think is ultimately what's at the heart of any 'Hydaelyn is secretly bad' theory; the developers introduced a big giant god, so inevitably people want to LB3 that thing in the face.
(Personally, I don't see the blue rock being an interesting boss fight, so I'll pass. But I'm also a healer main, so I don't exactly get to do the cathartic violence.)