I think that protestors should be careful about their tactics, but I don't think they should have to invent ways to tie a protest's theme to its objective. We protest when our moral values are offended, and that's enough of a reason.
But let's run with your notion. Suppose there is somebody who says: "I guess I didn't know about the civil rights violations and the deaths that this policy will result in. And I guess it's terrible that these refugees have been waiting years in limbo trying to rebuild their lives. But what's really important is that I expected to be home on time for dinner, so now I won't help them."
I don't think a person like that was every going to be sympathetic to the cause. They weren't going to do anything. They were going to ignore it.
And that's exactly the point of MLK's style of protest. It is to make a problem for some into a problem for all. It is to make it impossible for comfortable people to keep ignoring the problem.
So the solution is to harm more? That's not a peaceful protest, that's terrorism.
Raise awareness. Be loud. Be visible. But if you base your protest around screwing up someone else's day the response you get will more likely be "hey maybe you deserved it" than "wow we should help these people."
If you think you know a better way to lead protests, by all means show us how. But I have no time for armchair experts whose only contribution is "u r doin it rong bcuz I dont lik it".
The people who would say, "maybe those refugees deserved it" because something was mildly inconvenient for them were never, ever going to stir themselves to do the compassionate thing. Ditto the people who value a sliver of comfort over helping legal residents arbitrarily ejected from their homes.
Your claims here are those of the white moderates that MLK called the greatest stumbling block to black freedom. Apparently you haven't yet bothered to read his "Letter From A Birmingham Jail". But I'm done trying to spoon-feed it to you, so if you'd like to pursue this further, take it up with him.
Yes - some people might be less sympathetic, those are not the people the protest set out to influence. It's the people who haven't made up their minds yet, the people who think that "This will all blow over soon" and want to get on with their lives in their bubble. They NEED to be inconvenienced to react, to make up their mind that this is something they can't ignore. Only then when you engage the apathetic can the protest grow, and protests must grow otherwise most of them fizzle.
The problem with this approach is that unless you do it in a way that's relevant to the original issue, you're just dragging innocent people into your fight and victimizing them. Depending on how much of an asshat you are to them (like, say, stranding them for several hours at an airport when they've just got off a flight), there's a good chance you'll push them to be less sympathetic to your cause.