I got the first fairy already, I guess I just assumed that "Magical, mystical spring up in the mountains" (mentioned in a memory cutscene from Zelda, and some random person in Hateno Village who gives you a sidequest) referred to a Great Fairy (who generally are associated with springs).
I guess the weirdest thing about it is the lack of reward. Your reward for doing all that work is basically a new Goddess Idol location (in case teleporting up a mountain with damaging cold weather is better than teleporting to Impa's village, I guess), a Shrine (Spirit orbs are cool, I guess), and an Ice Spear in that Shrine (it's not bad, 20 damage but starting to become obsolete once I got into the Zora domain, which is literally the next step in the game).
I put in a lot of hard work to get up this mountain, had a sort of miniboss with the corrupted dragon, and got...basically a spirit orb, of which I have probably collected close to 30 by now for a lot less work.
It feels really odd. BOTW has been really good so far at rewarding you for your hard work, even when you don't know if you're going to be rewarded. Instead, this was kinda the opposite: Put in a lot of work for a really obvious reward, and got almost nothing for it. I can expect that from a lot of other games (Witcher 3 comes to mind, sometimes), but it's the first time I've experienced it with BotW.