Rock bands only get tourism from the civ they play in (unless you have the AoE upgrade), so wiping out the weakest civ would do very little.
It depends on how much tourism they've accumulated already. The AI isn't smart as the human in concentrating them vs the highest domestic tourist civ. I'm also assuming the human is the domestic tourist leader and is smart enough to run Music Censorship. And if not, even the AI is smart enough to do that once they get the chance.
We only target the weakest civ
s because they are easier to kill and killing them removes already established tourists, as well as preventing future tourists from them, whereas attacking the civ directly or stealing their great works will only affect the future tourists earning potential.
I suppose it is more accurate to say to target the weakest civs that they've accumulated the most tourists in but that's hard to guess and I don't know how to get the info, nor would it be worth the effort anyways. Perhaps if you notice where they've been sending the RBs too, you could wipe out that civ or simply raze cities with many wonders which will limit the places Rock Bands can play.
And finally I was meaning to compare it to methods like spying and trying to kill them. Spying isn't going to do much because stealing some great works isn't that much tourism plus there are almost always not enough spaces as opposed to great people so it can be replaced fairly easily. Not to mention you need your own space to hold stolen works too.
None of the other missions will do anything other than indirect damage. Killing them will surely work but is significantly more tedious and harder while probably distracting you from whatever win condition you had plan than just picking on the easy fruit and depending on how close the cultural game is, may just pass the problem to somewhere else, whereas wiping out some civs will hurt all CV players.
To be fair either method needs to be done a bit in advance. But the goal is just to delay them so that they can't overcome you anyways. Maybe just wipe out some civs in advance just to be safe?