Weird conflicted feeling..... :)
I agree with you on the outset, (assuming I understand correctly...) that for example slowing traffic at already busy intersections, at busy times etc does have the effect of getting more people to see a message at the cost of inconveniencing people who perhaps are already on the side of the protesters or as you say have nothing to do with whatever the protest is about...
I don't think that cyber attacks are a great deal different. Let's make up a company, say Sarny, they make um... sarnies. For some reason many people who are savvy online don't care for the actions of Sarny and use a denial of service or something to make their online presence go slow which is their form of protest. In comes another consumer who doesn't know/care about Sarny's online activities, but they need to connect to the same website to get some kind of update or information, firmware, whatever.
In an ideal world (IDK is it the ideal outcome?) - people who would not know about the reasons behind the protest, become informed, and then join the protest, at which point above company realises the error of it's ways and consults with the community at large to make sure they can make everyone happy. (I know I know, not the usual outcome, but we can still dream). The upsides being that both the consumer is informed of an issue they may not have known about before, and the company realises that either intentionally or not, they need to do something to keep their customers.
Now, there are many many other outcomes we could invent, imagine or point to previous examples of, but one that springs to my mind (one I can't answer but would be interested to hear what people think), let's say that DDoS became a valid form of protest. How much of the population at large would be able to really understand what it means and what the ramifications are? What if the online protest seriously damaged the company to the point of causing great loss or even bankruptcy? How does the law deal with extended in person protests that might damage a company still trading?
I know there have been times my primitive brain would have said 'yay 'x' company went down to people power', but I do wonder what happens to workers, communities, and .... shareholders (*shudder* ;)
Anyway just some ramblings