"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.

It sounds wrong to this British ear.

"There was a problem at the mill so I wrote them" - this sounds wrong

"The mill workers were complaining so I wrote them a letter" - this is fine

"The mill workers were complaining so I wrote to them" - this is also fine

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?"

Carol: "I wrote them."