...it's not live. David Crosby is a mile from his mic but we clearly hear his voice.
The tambourine is very prominent, but Gene Clark is not playing what we hear.
There are no amps or monitors, so nobody would be able to hear what they were playing.
Then, it fades out at the end.
Typical TV stuff in the 60s & some of the 70s. The TV people didn't want to have to focus on audio balance for a TV show, so they just spun the record & told the bands to pretend they were playing.
Also, some of the people didn't take rock seriously and figured the bands weren't really that good.
Weird, since the Beatles & Stones played US & British TV & they WERE live.
It went so far that some record companies took a B cut of the song to give to the TV shows so the "live" version wasn't identical to the hit version, though that was uncommon. The Doors did it, the Zombies did it, & a few others.
Oh and I'm with you on jangly guitar. So much so that I own three 12 string guutars!