"ASI is (formally speaking) a syntactic error correction procedure."
Yet the "expectation of ASI" or (I think more likely) "expectation of newline significance" makes people believe that they'll get a ; inserted by separating two things by one or more newlines.
Most languages do not specify error correction. HTML5 of course does; CSS too; among general programming languages it's much less common. The spec for ASI does not fit in the tried-and-true LR(1) formalism used by ECMA-262. Parsing is not all ad-hoc or equally well-formalized and proven.
In addition to ASI, ECMA-262 has to use lookahead restrictions and a bit of semantic checking to cope with what could be purely syntactic concerns (say, if it could use GLR instead of LR(1)).
The error-correction view seems to want to add a third category, strings that are in some sense "errors", but nonetheless get unambiguously mapped to an AST. Which is strange from a classical formal-languages view, because if a string gets mapped to an AST, it's in the language, and the procedure that mapped it constitutes the parser! That category seems more like "warnings" to me, i.e. you probably shouldn't do this, but it will produce a program if you do.
But that doesn't alter ASI's error-correction nature, which is not an "implementation detail" -- it's in the spec and all too observable.
You're right, it has the character of a warning system, like Dart's unsound "types". But if it had been noisy (consoles in the early days were costly), too many developers would have ignored the warnings, and users would have paid for the overhead.
Your concluding sentence is spot on, I agree people should use semicolons in JS. Relying on a Ruby-like coding style in the large is way too risky.