I think that's the right attitude. The point is to communicate clearly and in a powerful manner—for everyone. You can have the most Native accent and yet communicate poorly. Or you can have a foreign accent but work on aspects of it to communicate clearly. At the same time, Native speakers, too, should work on their comprehension skills, especially in increasingly diverse societies.
Haters gonna hate. But you don't even need them. If you speak clearly and proudly, those who need to understand, will.
Regarding the 2013 study on British/working-class New York accents: Actually I believe your description exactly matches the one in the article, which says, participants had "higher false memory rate for the speaker with a working class accent," suggesting they thought that voice was making a mistake, whereas with the British voice they thought it wasn't a mistake and that "the Brit must have said what he meant."
In general, there are two effects at work. One is stereotypes: cultural bias, which led the student in Rubin and Babel's studies to just assume that they won't understand the accent of a Chinese looking professor. Another cultural effect is that high-prestige accents like the British one in Lev-Ari study, are held more reliable.
The second effect is cognitive: we have a tendency to avoid actually listening and paying attention to difficult stimuli, in this case, foreign accents.
These effects are not mutually exclusive. And they also overlap. Stereotypes are partly born out of lazy brains. But they also shape how our brains interprets the world.
...but the author is using it to support the point that "...we slip in what we think a foreign speaker means to say, particularly when their accent has low social standing." Which seems like it should mean that the participants would have a higher false memory rate with the British speaker.
It's an interesting question, though, whether a working-class New York accent is less "foreign" than an upper-class British accent to a Stanford University student.
Maybe you are not interpreting the term "false memory" in the right way: Here, false memory means that they falsely remembered something that didn't actually exist (a word that the New Yorker had not said). It's only a term used in the study to "describe" their action. In reality, they perfectly remembered that the New Yorker had not said a word that was previously on the list. But they assumed it was the New Yorker's "mistake." With the Brit, they assume it was his "intention" to drop a word.
Also, another thing is confused here and you are right. For this: "...we slip in what we think a foreign speaker means to say, particularly when their accent has low social standing."
The study is used to support specifically the latter part, that is, bias against accents with low social standing. Not foreignness. (An accent can be native but have low social status, it can be foreign AND low social status, or it can be foreign like the British one but have high social status.)
