Go for it. I'm not ashamed of being wrong Could Mohammed and the early successors write?
Mohammed was not "schooled". The "iqra!" phrase could refer to reading from text, or reciting from memory.
Many of his companions were literate though (as in they were scribes etc, or knew how to read Aramaic). Umar (3rd caliph) was literate...his sister was literate before she converted, I could give many more.
So you thought some English lady went into tents and transcribed bedoin lore or something? LOL...Baghdad was the center of Arabic literature and it was very literate, like a benchmark in its heyday.
Alf-Layl wa layla were mostly compiled 400-500 years after Mohammed SAAW's death, in golden age Baghdad and had a lot of Persian and Byzantine input. "Arabian Nights" was a cheesy B-rate movie type choice for a name. English is overrated.