Every time Link takes four climbing "steps"
in a row in rain, he slips.
Every time you jump, he slips (though IIRC the fall is slightly less than the height of the jump?).
Take three "steps," then jump. You'll still slip after the jump, but you'll still be much higher than where you started. Once you get the 1, 2, 3, JUMP rhythm down, it become second nature to switch to it when it starts to rain.
Once you combine this with:
-Climbing gear (especially the level 2 gear that reduces the stamina cost of climb jumping)
-Stamina upgrades and potions for the most daunting and tallest peaks
-Realizing just how generous the game is with allowing Link to stand on inclines to rest for a couple seconds to fully recharge stamina (seriously, he's practically a mountain goat or a Skyrim horse)
Then it becomes trivial to climb up things, even in downpours. I realized the rhythm very early on in my first playthrough when I tried to climb Dueling Peaks and it started to rain. I made it up the entire damn mountain anyway.
Revali's Gale is also obviously the final piece in that it allows you to outright fly over small to medium sized cliffs, and have a drastically higher starting point for higher ones, but it's more of a "yeah, this is trivial now" than a necessity. Once you have all of those, there's like... maybe like two or three outright sheer cliffs where rain would maybe make it almost "impossible" without constant chugging of potions, and in those cases the same would be true if it weren't raining.
