Actually they did write some songs and all of them could play.
They certainly could, Nesmith in particular. The irony is that none of them were on TV playing their primary instruments; for example, Davy Jones was a drummer, not a front man.
There was, of course, another David Jones in music at the time who had to change his stage name because the Monkees one was already taken. David Bowie.
Yes, they wrong some songs but they weren’t allowed to record their own songs until later in their career, when they weren’t really scoring many hits.
And I believe only two of them could really play instruments at the start, Tork and Nesmith. Both Dolenz and Jones were actors by trade, and though they had singing experience, they were not musicians when they were recruited.