There are plenty of ways this can go wrong. The little weights Gaudí used might not correctly model the weight of the stone. The whole thing might be unstable under inevitable sideways forces from wind and such, although one could blow on the string model to make sure it doesn’t start swinging to approximate this. And the string model probably doesn’t say much about the distribution of compression loads in the stone, especially if the overall structure is not statically determinate.
Imagine a single weight dangling from two strings of roughly equal lengths attached near other on the ceiling. The weight will swing if you blow on it. If you turn the strings into (lightweight) compression elements and turn the whole thing upside down, it will be a bipod. The weight will indeed be balanced, but any slight perturbation will knock it over or will at least require shear forces to avoid this fate.
Now try this with three strings. The weight will resist swinging if you blow on it, and the cathedralized version is a stable tripod.
Now try four strings. If the string model is imperfect, one string might be slack. If you cathedralize it, you get a quadropod, and, if the construction is at all imperfect, the loads on the four legs might be wildly different and, if it’s too far off, the structure might wobble. This structure is statically indeterminate.
There is, strictly speaking, nothing strictly wrong with static indeterminacy, but it makes the analysis considerably more complicated.
Another issue is that Gaudí used a bunch of little weights instead of weighted strings. This means that he’s modeling stone columns with all the weight concentrated in a few places, which isn’t quite right.
This is only really relevant to cases where the loads are higher than the weight of the arch/chain (manmade arches typically are just symmetrical as the loads are far less than the arch's).
I'm sure someone here is capable of elegantly describing these loads in the language of math to describe the same conclusion. That's not really the why part, just a different language for explaining it.
In a hanging chain, the sum of the forces pulling down (due to gravity/load) and away from the end (due to tension) on a given link must point in a direction exactly opposite the angle of the next chain link toward the end. Else that chain link would rotate until the above is true.
In a stone arch, the sum of the forces pushing down (due to gravity/load) and toward the base (due to compression) on a given stone must point in a direction exactly equal to the next stone toward the base. Else that stone would rotate and the arch would break.