I know of both the over-strike method for traditional typewriters and extended ASCII. But I have no knowlede of a ASCII (extended or not) supporting every sign commonly used in English. Granted, it is not as significant as other languages missing whole letters, but signs like ”,“,—,–,’, which are often replaced with simpler ASCII alternative (",') or their combination (---,--).
Over-strike, from what I remember wasn’t implemented in any well-spread way with standalone ASCII, although I could be simply not aware of it. It gives us then a few more characters rarely used in English language in loanwords (née, naïve, façade).
Skipping diacritics with uppercase letters on displays with too little space happens with Polish language too, so I am aware of the practice.
In any case, yes, overstrike is just an approximation, but believe it or not, ASCII was designed to make it possible. And yes, it kinda sucks, and yes, it doesn't really make ASCII a multi-byte encoding, not exactly. But it's a funny thought that ASCII kinda almost was a multi-byte encoding. One can imagine ASCII evolving to treat BS<mark> as not unlike Unicode combining codepoints to make it possible to get proper diacritics even on capital letters, but ASCII would still have been a dead end, naturally.
The comment you're replying to just said ASCII without qualification. By definition this is not multibyte, or even whole byte (assuming by byte you mean octet). ASCII is a 7 bit standard, anything else is some other encoding.
I'm quite certain that ASCII was designed to make overstrike feasible. Overstrike just wasn't explicitly part of the standard. The quip about it being a "multi-byte encoding for Latin" was a joke made funnier (to me, and perhaps only to me) by having a kernel of truth in it.