UPDATE: someone on Discord reminded me of a loose end in Kakariko. I think that will solve my issue.
Here's what I did:
I found Mineru on my own after having found the construct factory, even though I was nearly blind in the storm on Thunderhead Isles.
I'm still missing one of the Hylian stone tablets, so I tried exploring more of Thunderhead Isles but got super frustrated with the lack of visibility. "Surely they don't mean for me to be here!"
I searched the net for "Is it possible to clear the storm...", and determined it is, but I have to finish "Crisis at Hyrule Castle", which I was putting off because I thought it would end the game.
I try to finish Crisis at Hyrule Castle, but I didn't realize the quest marker jumped up and down the levels of the castle so I just went down, down, down thinking it was a generic quest marker in the center of the castle, and not the next spot to find Fake Zelda.
I find the Demon King and beat the game.
When I searched for answers on the net, it said I would receive a quest called "The Fifth Sage" and the storm would be cleared for me, or I would have some means of clearing it myself. (I didn't read too closely since I was trying to avoid spoilers.)
After beating the game, I went back and tried to finish Crisis at Hyrule Castle again, and I did. Fake Zelda kept jumping around, and I went to each spot. And, damn! I had missed an entire chunk of the plot where I face the Demon King for the first time with the 4 sages, and he reveals that Fake Zelda was indeed Fake Zelda, and there's a big fight and I get a heart container.
When I went back to Purah to turn in the quest, everyone's like, "Wait, you already found the fifth sage, and the Master Sword? Why didn't you tell us!?" No talk at all about the storm on the Thunderhead Isles. I received and then immediately finished the quest named The Fifth Sage, and my only quest now is "Defeat the Demon King".
Can I clear the storm so I can properly explore the Thunderhead Isles, or is my game just borked?
I can understand Nintendo not locking Mineru behind some arbitrary plot lock, but it kinda sucks that I did things out of order and was able to skip entire sections of plot and beat the game by accident. The entire time I literally thought some plot device would boot me out and I wasn't about to beat the game. (It became obvious when I could no longer fast travel out, and I just went with it.)
What was supposed to happen? Was I supposed to be blocked from fighting the Demon King until I found Mineru? Was she needed to advance? I feel like once you're in the castle, you're going to discover the chasm and go exploring. Is this game similar to BotW where there's nothing stopping you from killing Ganon as soon as you leave the Great Plateau? Or do you need to have all 4 (or 5) sages in TotK to beat this game? Just curious.