I agree with your assessment. As you and Adam suggested, there is some overlap between the way a strikethrough is used and epanorthosis. But I think there are significant differences, so I would not apply the term "epanorthosis" as a general label for the intentional use of the strikethrough, as suggested by comments to the original post referenced.
1) First of all, here are the ways in which epanorthosis is used:
a) To make a legitimate, honest correction or amendment to something that was stated previously. It is obviously used in verbal communication, and is also used in writing when depicting speech or stream of consciousness.
b) To pretend to make a correction for humorous effect or deliberate exaggeration. "Sigmund Fraud--I mean, Freud" is an example of this. So is something like "millions--no, billions!"
A strikethrough can also be used for humorous effect, as in 1b above. However, I would argue that it can't quite be used as in 1a above. Online and in printed documents, if the writer can go back to the text to apply the strikethrough effect, they can also go back and delete it altogether. So no one can claim that they were making a legitimate correction. Nor is a strikethrough a good way to imitate conversation or stream of consciousness, because it inherently implies that the writer is able to either modify what's been said after the fact, or correct it before it happens by setting the font. Obviously neither one is available for spontaneous speech or thought.
A strikethrough can be used for legitimate correction in a written document, when the writer doesn't intend to make a clean copy, but in that case it's really used as editing markup, and thus is not part of the text itself. It's truly not intended to be read, and while someone may still read it, what they're reading is more of a meta-text--traces of the process that led to the final version of the actual text. Online, the strikethrough font can also be used to display document edits, and in this case it's also only part of a meta-text--the documentation of the revisions made in past versions to form the current text. But it's not the same as the deliberate, textual use of the strikethrough. Even the syntax of HTML5 makes this distinction: there is a different tag for the strikethrough effect ("s") and the effect intended for document edits ("del"), even though both are usually rendered by the strikethrough font (https://developer.mozilla.org/en-US/docs/Web/HTML/Element/s).
(But see 1006a's comments below for a more nuanced consideration of interactive sites.)
2) So here are the ways in which a strikethrough can be used:
a) as editing markup, described above.
b) as epanorthosis: as in "Sigmund
Fraud Freud," or " millions billions." The type of comedic effect is slightly different here than in the case of a written correction, because the reader is given the "punchline" right away, as you pointed out. The dramatic effect is also lessened for the same reason. But it's still the same general technique.
c) to create an internal monologue/inner voice effect alongside the speaker's 'official' voice, as in your example: "I went to the store and
I just realized I forgot pickles saw my friend." You already described in your question the reason why this font is so effective for this effect: "the reader is forced to create a second train of thought."
d) as part of a paralipsis: "Stating and drawing attention to something in the very act of pretending to pass it over" (http://rhetoric.byu.edu/Figures/P/paralipsis.htm). For example: "Jane is a very dedicated employee and I won't say anything bad about her
like the fact that she steals." This example would be a paralipsis even without the strikethrough, but the font emphasizes the effect. Here is an example where the paralipsis relies entirely on the strikethrough: "Jane likes her five-finger employee discount." If the strikethrough wasn't there, the sentence would just be an outright criticism instead of an ironic jab.
c) signaling allofunctional implicature: when a sentence has a different function than the one implied by its grammar (https://en.wikipedia.org/wiki/Sentence_function#Allofunctional_implicature). For example: "let's take all the artificial ingredients out of twinkies:
chemical1, chemical2, chemical3, chemical4, chemical5." Here, by using visual effect, the markup changes the sentence from imperative to declarative ("there is nothing but artificial ingredients"). Another example is " Does anyone like this?" The sentence is changed from interrogative to declarative.
d) as typographic emphasis for something bad or deprecated (anti-emphasis?). For example: "do this,
e) as sarcastic negation: For example: "
this is so much fun." The strikethrough negates the statement, but not in the same way as saying "This is not fun."
f) to signal ambivalence or passive-aggressiveness. For example: "
I hate you." Obviously if the writer didn't mean it, or truly didn't want it to be known, they would have erased it altogether. But if they mean to be aggressive, why cross it out?
And there may be other uses as well, of which I'm not aware.
So the concept of epanorthosis and strikethrough usage intersect only in cases 1b and 2b above. They are neither equal nor subsets of each other. One of the things that a strikethrough can be used for is one type of epanorthosis.