I wouldn't say narcissistic, but socially anxious people do tend to be very self-centered and some even think of other people's problems as unimportant or non-existent. There's this bizarre notion that narcissists are the polar opposite of SA, promoted by Youtube channels that take advantage of people's insecurities and convince them that anyone they don't like is a secret narcissist and that they're doing the right thing by trusting their instincts and avoiding everyone. But I think both disorders have a lot more in common than we'd care to admit: sensitivity to criticism, taking rejection personally, being upset when people don't automatically behave in a way that benefits you, and in some extreme cases thinking you're better than extroverts and masquerading any rejection as part of your four-dimensional chess game.
I think you make some very good points.
I've always thought that most people would be very focused on their own problems if every day they're forced to deal with severe anxiety. When pretty much every part of their lives is affected by anxiety - and they have to try and find a way to deal with all the various situations that life entails. Things that other people can do without giving it any thought.
I can see how many people would interpret that as self-centredness when really it's more a case of survival.