They all could play and sing before The Monkeys. It was just that the instruments that some of them played were not necessarily the instruments that their characters in the show were supposed to play.
Davy Jones was a singer before The Monkeys. He had a single make the Billboard Hot 100 in 1965, and a couple singles that made the top 10 that year in Australia. He could also play drums. But he was also short and it wouldn't work out well visually to have him on drums.
Micky Dolenz had a band of his own in the early '60s where he played guitar and was lead singer, and also wrote songs. But his character in the show played the drums, which he had never played.
Mike Nesmith played guitar and bass and sang in several folk and country bands before The Monkeys, and wrote songs. His song "Different Drum", written a year or two before The Monkeys, became a hit for Linda Ronstadt in 1967. He had offered that song to the producers of The Monkeys but they didn't want it.
Peter Tork played piano, base, guitar, and banjo long before The Monkeys and had spent a lot of time doing folk music in New York.
They were allowed to have some songs the wrote on their albums. The first two albums each included one Nesmith song and one song Nesmith cowrote.
Their 3rd included 3 Nesmith songs, 2 songs that they all wrote, a Dolenz song, a song Tork cowrote, and one that all 4 cowrote with one other person.
Their 4th included a Nesmith song, a Tork song, one cowritten by Nesmith, and one cowritten by Jones.
Their 5th included 3 by Nesmith, and one he cowrote, and 2 that Jones cowrote.
It's not that the Monkees 'couldn't play' - that show biz act was selected for visual appeal while management had the Wrecking Crew studio a list create the pop songs. It was brilliantly successful business.
The Beatles were arguably an earlier incarnation of this - a v rough and ready covers band propelled to 'stardom' by Epstein and heavily produced, notably by George Martin. Just like Jimmy Miller's shaping of the Rolling Stones in their glory years it's the people behind the curtain who transition these acts into the big league.
(The Jones Stones were a very good UK R&B covers band in their early years just as the Beatles were originally a rock and roll covers band)
All bands need help in one form or another. The Beatles understood this, and would get help as needed (even Eric Clapton contributed). Michael Jackson got Quincy Jones to produce his breakout albums. The Monkees believed they didn't need help, and could do it all on their own. "Headquarters" was the result. It's a very weak album.
In my not-so-humble opinion, the Monkees were given the greatest opportunity dropped in their lap, and completely muffed it. They just weren't smart enough to compete with The Beatles.
The thing is the Beatles actually wrote their material - the Monkees sang tin pan alley tunes written by Carole King et al and played by the wrecking crew. They never had management or support to do anything creative themselves except mug for the cameras. Show biz vs musicians...
They did launch a lot of careers though, like Louie Shelton's https://youtu.be/7L6Vin-5188
