I have a huge collection of music, my favorite stuff largely being from the mid to late 60s. There were indeed thousands upon thousands of political rock songs at the time, most of them being obscure small sellers that didn’t chart. (Most of my collection are love and drug and party songs that didn’t chart either; I don’t think the lyrical content had much to do with this).
The first rock-not-folk song I have that’s overtly political wasn’t about the Vietnam War, but nuclear war. I recall it dates from 1965, titled “Answers Please,” and was by an Irish band that I can’t recall the name of (can’t be bothered to dig for it at the moment, sorry). They sound a bit like Them. You’d be surprised how many anti-nuke war songs date back to the late 60s. Actually some Joe Meek-written stuff takes us back to the mid-60s. (And then we have the ‘fun’ side of nuke war, Bill Haley’s “13 Women,” the first nuke war refrerence I know of in rock’n’roll: “There were 13 women and only one man in town…”)
Other obvious songs not metioned yet would be “Monster” and “Ostrich” by Steppenwolf, a lot of MC5 material (how could Alice Cooper not recall this? they were Michigan contemporaries), a few Spirit songs, The Fugs, The Monks (“Complication”), the list could go on forever. Even the insular, decadent Velvet Underground notes “all the bodies pilin’ up in Nam” in “Heroin.” Even The Monkees wrote an anti-war song called “War Games.” Then you have the increasingly overt racial & other social messages of Sly & the Family Stone, Curtis Mayfield, eventually the Last Poets…
If we count generic ‘pro-peace’ tunes of course the # skyrockets even more. Include pro-drug statements as political… whew!
I have a handful of pro-Vietnam war effort songs, most of them quite obscure (“Christmas in Vietnam,” anyone?) but even one by Jan and Dean! I have A-side of a 45 on a recently compiled comilation by one fellow who comes out for the war on one side and - unfortunately this didn’t make the comp - against mini-skirts on the other. The compiler wrote in the liner notes that the singer “couldn’t find the right side of any issue.” 
I was born in 1971, so I don’t have recollection of this, but listening to any music of the era I think the listener has to remember that the whole psychedlic rock thing marked the listener as being on one side of a cultural struggle with political implications, thus in a liberal (excuse pun) interpretation “Are You Experienced” and “Rain” or even “Interstellar Overdrive” drew a line n the sand in a way a song which isn’t overtly political couldn’t do today. We (people to young to have been around for the original) are just used to rock as being ossified and institutional.
I can’t stress this enough, but the impact of this new politicized youth culture was GLOBAL. I have pro-peace psych rock songs from around 60 countries on every continent, from Japan to Peru to Iceland to South Africa… even a Czech song in English that must’ve slipped past the authorities (“Stand Up and Go” by The Beatmen.)
The initial reaction to institutional rock was of course punk, and that’s where I have to disagree about political rock tapering off after the Vietnam War. It exploded in punk in the 80s, although almost all of the releases were tiny pressings that had no chart impact. (Most of the songs we think of as being 60s iconic didn’t actually chart themselves; some weren’t even released as singles.) There were probably more political punk songs released in 1980-1989 than there were rock songs of any sort 1960-1969, considering how technological changes dropped the cost of releasing material. By 1989 the average garage punk band could have a few albums, while the 60s equivalent band was lucky to get out a couple of 45s.
) but I think rock and politics are inextricably linked. I’m just old enough to remember the real protest sonds. Even before that, there were songs like “Summertime Blues”, that, while not exactly political, did have an element of protest.