Yes, very important. I’m surprised Rav Moshe didn’t think of this.
That’s why I shake hands with women. In certain situations I have no choice but to hug them also.
I also frequently eat at my goyish neighbor’s house. I don’t want them to think I’m some kind of nut case. Even worse they might be insulted. And we all know that is against The Torah!
I suppose I could figure out some way to explain to women why I can’t shake their hand; or explain to my neighbor why I can’t eat his food. But nah, they still might think I’m lacking Derech eretz.
Besides bacon is delicious.