I have english as my first language but I moved to a country where most people learn it as a second language and people are horrified to see how I communicate with my sister. Of course, I wouldn't do it with them when i'm actually trying to communicate effectively. I think its one of those "when you understand the rules well you can break them" type thing, here it indicates a carefree and playful tone, for a similar reason to how the lack of capitalisation in the title etc indicates a casual tone. It's trying to avoid being Official and Boastful and Buisnesslike.

In particular, the first sentence is similarly playful with its choice of words; suffice to say this is intentionally poetry

It’s still an adverb here. The meme used here is to deliberately the verb as a form of wordplay.

Would it help to still consider "accidentally" as an adverb, but modifying an omitted verb? As a native English-speaker, that's how I tend to parse it.

Up until the end of the sentence, the reader doesn't know whether a verb is going to occur. English usually arranges a sentence with subject-verb-object ordering ("I ate my meal."), and not subject-object-verb ordering ("I my meal ate."), so this is unlikely. However, variations occur, such as "Neither a borrower nor a lender be." in Macbeth, or "Frankincense to offer have I" from We Three Kings. These tend to be somewhat archaic usage, and so an author may deliberately use this tone for a sense of formality.

This is similar to a garden-path sentence [0], where the sentence cannot be easily parsed in order. Up until the title ends, we don't know if the title will so formal that it changes up the word ordering (e.g. "I accidentally a scheme brought forth.") or so informal that it omitted a word without realizing it. This contrast between extreme formality and extreme informality becomes a source of humor when resolved.

(And by dissecting a joke, I have made it that much funnier. I'm great at parties.)

[0] https://en.wikipedia.org/wiki/Garden-path_sentence

I am french I read your comment then read again the title to see what was wrong and I noticed my brain automatically added the word "entered" after accidentally. So yes I understand your feeling

I don’t see what ESL has got to do with that preference.