Since so many people struggle with this dang puzzle and most of the online guides are vague at best, I decided to find a solution that would always work, if set up correctly. You do not need any glitches, exploits, cheats or special skills - just an iron sledgehammer, stasis, a cube bomb and about two minutes to set it all up.
Step 1:
Hit the switch to make a shrine orb appear. Stand behind it, target the orb with ZL and summon a cube bomb. You need to place the bomb on the ground behind the orb, so that the crack between the two panels is approximately in the center of the bomb (see screenshot). This is the correct distance, but not yet the correct alignment.
Step 2:
Pick up the bomb again, then target the orb again. Carefully move the camera until the "hoop" on top of the bomb and the bouncing orange "target" arrow align - this is the direction in which the orb will fly off. If you move Link left/right you can fine tune the trajectory to make the orb fly off precisely where you want it to.
You want to aim for the center between the two last pillars, or slightly to the right. This is the right direction, and if the bomb is still at the right distance once you set it down (center of the crack) then the actual setup is complete.
Step 3:
Go to the OTHER side - facing AWAY from the goal, and stasis the orb. Whack it with the sledgehammer 4 times. I usually use a spin attack as it is faster than hitting the orb manually, but you need to cancel the spin with B then (no "ground slam" at the end).
It does NOT matter that the trajectory arrow is pointing in the wrong direction. You only need to "charge" the orb with the right amount of force.
Before Stasis runs out, move away so you're out of the blast radius, and detonate the bomb.
Step 4:
Watch the orb fly into the hole. You did it. Go to the treasure chest, pick up your well-earned Giant Ancient Core and revel in the fact that you won't have to engage with this finnicky trial-and-error design failure of a puzzle ever again (unless you start a new playthrough of course).
In case you'd rather follow a video tutorial explaining the steps, here you go: https://www.youtube.com/watch?v=E_-VpeTmuCs