"For such a small sample, that's not enough to definitively say whether the small difference is due to random chance, or due to small differences in opinion in the population being surveyed."
Then you haven't shown anything. Without statistically significant data, your survey is meaningless.
"What it does show, even with such a small sample, is that in the underlying population there's almost certainly no huge gap between people's opinions of black women vs. white women breastfeeding in photos."
No, it doesn't. You cannot draw conclusions from your results without significant data, because as you just said, your results could be due to random chance. I see this all the time in papers submitted for peer review. They'll say something like, "our technique showed benefit over the other techniques, even though the difference was not significant", and try to claim this as a win.