That has to be a mistake; there's be no point in supporting those addresses unless they were writable.
I took a look in the Model III Technical Reference manual and couldn't see where it says that. Do you have a page #? I'd like to note it as a correction.
FWIW, this contention that they're read only, not write-able, appeared in several 80 Micro articles. But... yeah, I never checked it.
Here's the exact text about this from the Model III tech manual:
It only mentions reads, not writes, but it says what part of the schematic to look at, so...
Looking at figure 8 there's a 74LS30 8-input NAND that takes massaged input from 10 address lines (A0 isn't one of them, IE, it can't tell the difference between 37E8 and E9) and ROMCs and turns them into an "LPEN" signal. This then goes to a 74LS32 that ORS it with RD, which is a memory read, and *then* this signal is AND'ed with the "LPIN" signal from the port decoding. So... yes. Unless I'm missing something this decoding isn't going to be satisfied on a write, it's *only* going to work on a read. (IE, the transaction being a read is a specific requirement of the signal generation to activate the port to read, and there's no provision there generating the counterpart "LP OUT" signal from a memory write.)
You're definitely correct that it doesn't make a ton of sense they didn't preserve that address as writable, but I'd just chalk it up as another entry on the list of kinda questionable compatibility breaks they made with the Model III. I think the justification was that Level II BASIC didn't actually have a command to read printer status so PEEKing the status address was SOP even if you were using LPRINT for the output.