I was looking at the "protest" picture, and it strikes me as odd that there are so many nice and neat Spock signs. Was the Vulcan that much of a phenom, then?
Ohh, absolutely. Spock was the runaway star of TOS, especially among female viewers. That's why the second and third seasons both opened with Spock-centric episodes -- because the network was getting huge positive response to Spock and wanted to play him up as much as possible. Read the behind-the-scenes book and you'll see that Roddenberry and Shatner struggled with the ascendance of Spock as the breakout star and the effort to keep Kirk central to the show despite that.
I have the vibe that this was a little staged - it looks like a protest scene from an episode of Batman!
Definitely. The supposedly spontaneous fan protests and letter-writing campaigns were (not so) secretly organized by Roddenberry himself.