If I recall correctly, Mickey Dolenz could play the guitar and auditioned for the show with that instrument. But as you point out, the producers wanted a drummer and not another guitarist so Mickey had to learn to play the drums.
Boyce & Hart deserve the credit for the Monkees music. B & H wrote the theme and most of their hits. A lot of studio musicians made the early hits sound good.
The Monkees eventually did perform the music themselves and I think they won the respect of other musicians.
At least they weren’t autotuned.
I remember reading he disliked being named the drummer, since it put him in the back and not in the spotlight. But, in any case, he often sang lead (for “Last Train to Clarkesville” and “I’m a Believer,” for instance); Tork and Nesmith thought his voice gave the group a distinctive sound.
I’d say they wrote probably a quarter to a third of their most popular tracks.
- Boyce and Hart wrote “Last Train to Clarksville”, “I’m Not Your Steppin’ Stone”, “I Wanna Be Free”, “Words”
“Valleri”, and “Tear Drop City” - Neil Diamond wrote “I’m a Believer” and “A Little Bit Me, a Little Bit You”
- Micky Dolenz wrote “Randy Scouse Git”
- Goffin and King wrote “Pleasant Valley Sunday” and “Porpoise Song”
- John Stewart wrote “Daydream Believer”
- Leiber and Stoller wrote “D. W. Washburn”
- Michael Nesmith wrote “Mary, Mary” and “Listen to the Band”
There was really only one album, Headquarters, on which they really played as a traditional rock band together. After that, Micky and Davy pretty much gave up playing instruments on their recordings. They went back to using mostly studio musicians, but more at their own direction rather than dictation by someone like Don Kirshner.
What a lot of people seem to forget is that up until the Beatles, it was very common for pop stars to only sing on their records and have the music done by studio musicians. How much guitar did Elvis really play, after all? The Beach Boys?
I don’t remember who said it, but I remember some wag giving Elvis credit for “teaching us all how to hold a guitar” 
One thing I don’t think the Monkees get enough credit for was how good of a singer Micky was. Davy was pretty good, but Micky’s voice was really just right for the kind of music they were doing.
A lot of people didn’t realize he was singing. Lots of people assumed Nesmith was singing because he was the lead guitar and it didn’t sound like Jones.
One of the more bizarre pairings on a concert tour was when Jimi Hendrix opened for the Monkees for several dates after the Monterrey pop festival. Especially when several months earlier Hendrix blasted the group as dishwater. Needless to say it did not go well as the teenyboopers kept chanting for Davy throughout Hendrix’s set.
Pretty much any TV ‘performance’ by a rock 'n roll band was lip-synched back then. Pull up practically any YouTube of some group on Ed Sullivan or Shindig or whatever, and if you’re familiar with the studio version, you’ll quickly notice that that’s what you’re listening to.
Here’s one of Eric Burdon and the Animals ‘performing’ House of the Rising Sun, for instance.
Often on TV appearances the vocalist would be singing live, but the instruments were all pre-recorded. One amusing variation was when Steppenwolf appeared on the Sullivan show. John Kay had a live mic, but he was singing over the the actual record, with the original vocal intact…so you got two John Kays singing in unison. Here’s a snippet.
That’s hilarious. Did no one notice?
I don’t know if this is really true, but I’ve been told that – since anyone who knew anything about the music industry in the '60s understood that bands “performing” on TV were mostly miming – it was considered cool for musicians to not take it too seriously and do things to make it obvious that the track was pre-recorded. I’ve seen clips of The Who performing on TV where you can hear the drums but drummer Keith Moon is on-camera throwing his drumsticks up in the air.
Sometimes the synch is slightly off. 
Bands often played live on the Smothers Brothers Show but for some reason I should probably look up, they decided to have the Jefferson Airplane lip-sync.* Grace Slick was so mad that she held up the plug from the unplugged amplifier and sang into it instead of a microphone. You can see it at the beginning of this YouTube clip.
*My memory says Smothers Brothers but I’m never surprised when my memories of the 60s turn out to be wrong.
Did these people who assumed it was Mike not watch the show? Micky is quite clearly shown singing (on the Micky songs). Sure, lipsyncing, whatever, point is there’d be no reason to think Micky was lipsyncing to Mike.
There’s a clip of the Mothers of Invention miming to “Son of Suzy Creamcheese” with Zappa at the microphone rapidly mouthing the word “motherfucker” over and over.
I assume they weren’t watching. It was a 1/2 hour show that ran for a couple of years. Far more people heard the music on the radio than from watching the show.
In fact, in some of the episodes, Micky would horse around while they were supposed to be miming.
Weren’t Peter Paul and Mary a created band? They weren’t a group of folkies who got stoned and decided to form a band. They had to audition for slots when a manager decided to create a new folk group.