The answer is that’s just the way English is. Exactly 1 is singular, everything else is plural (mostly).
“On accident” is American English, as a British English speaker I’d consider it a grammatical mistake. The same goes with “I forgot it at home” and similar constructs. However they’re correct American English.
I think the reason that "accident" is confusing is because of "I did it on purpose". As a fellow British English speaker, I would never say "by purpose!". By and large, I think that US English tends to be more logical.
"I wrote them" doesn't sound completely wrong to a British ear - it just gets misunderstood! I thought it sounded like exactly the correct way to say "I wrote the letters", until I got to the last couple of words in your post and had to reinterpret it. :-)
Quite, I struggled to formulate the example because indeed reading it the interpretation is that "them" could mean letters, and so doesn't sound completely as wrong as it does in the context of a person.
A better example would have been "I wrote Alice last week". Correct US English, utterly grating to British English. ( Technically still might not grate if your brain jumps to Alice being a Poem or other work of art! )
Because in British English we write letters, we don't write people. I don't know the term for it, it's not transitive vs intransitive, it's the verb object having a different restriction.
In my example, the "them" isn't referring to the recipients of the letters. It's referring to the letters themselves.
Mallory: "The mill received a series of letters. Those letters are evidence. Now, we're not leaving here until I find out who wrote those letters. Alice, did you write them? Bob, did you?"
"I left it at home" is common, but doesn't have the exact same meaning. Tbh, I don't think there really is a way to say that succinctly in British English—we would probably say "I left it at home", "I forgot to bring it", or—if the full meaning is strictly necessary—"I forgot it, it's at home".
Really, "I forgot it at home" is short for "I forgot to bring it; I left it at home".
“On accident” is American English, as a British English speaker I’d consider it a grammatical mistake. The same goes with “I forgot it at home” and similar constructs. However they’re correct American English.