Yeah, you want to quick cast the fireball as soon after turning as possible. Because the trick mainly works when you're close to the top of your jump arc, so turn + fireball need to be pretty quick after.

I don't find that stick is easier than dpad for the trick, but probably depends on the person. I personally wouldn't switch for just that. (If you do decide to switch to kbm, you'll probably want the SOCD Cleaner from resources to help clean up directional inputs.)

For the fireball skip, make sure you're using quick cast for the fireballs and practice jumping and turning at the peak of your jump and firing. You can practice this anywhere to get a feel for the rhythm, like by the spring in Crossroads where you can refill soul easily. Using the debug mod on PC also allows save states, which would help a lot for practice.

The main way I learned it was by going for it in every run (after some initial practice) and having a sensible backup on the fly below. Just got more consistent at it after a while.

I just checked the Downpatch link in Resources and it's letting me download from the mega. Might be an issue on your end. There's a Discord for Prey speedruns, too, if you want to hop in there and get some help.