Ooofff, and it even cost you multiple fairies. Those thunderstorms can be a real pain…

Spoiler: The earliest you can theoretically clear this section is directly after obtaining the “Regional Phenomena” quest, but you would have to find the right spot in the raging storm without any guidance (hint: map coordinates greatly help here) and then glide under the chamber that contains Mineru’s mask and ascend into it, to bypass the “10 heart container” door. You can also reach the chamber without even obtaining the Paraglider, but then the mask will say something like “now isn’t the right time” and won’t let you pick it up / progress the quest.

If you happen to have the Sheikah Sensor, there is a trackable shrine very close to the door. ;)