I have no strong intuition about whether this is true or not, but I do intuit that if it's true, the value of sufficiently for which it's true is so high it'd be nearly impossible to achieve it accidentally.

I'm not sure. The analogy might be similar to how an sufficiently complicated process is extremely likely to be able to model a Turing machine. .And in this sort of context, extremely simple systems do end up being Turing complete such as the Game of Life. As a rough rule of thumb from a programming perspective, once some language or scripting system has more than minimal capabilities, it will almost certainly be Turing equivalent.

I don't know how good an analogy this is, but if it is a good analogy, then one maybe should conclude the exact opposite of your intuition.