There are people with neurological deficits that make it harder for them to socialize, and there certainly are social skills (etiquette, tact) that some people are more proficient in than others, but I don't think "lack of social skills" and "social anxiety" are connected.
Ever watched other people socialize? A lot of them are terrible at it. They're rude, obnoxious, annoying, offensive, combative, etc.; but they don't have social anxiety. My sisters are terrifically bad at interacting with other people and utterly fearless socializers. Their fearlessness is part of what makes them so terrible. They still manage to get jobs, bfs, friends, etc., despite making every other person they meet angry at them. I happen to have pretty good social skills but you'd never know it because I go out of my way to avoid interacting with people.
Anxiety has nothing to do with whether or not you know what to say or do in a social situation. You can know exactly what to say and be afraid to say it. You can say the worst thing possible and not care how many people you're offending. Anxiety can distress you to the point where you can't think of what to say or do even though you know it, and it can lead you to self-isolate and therefore remain ignorant of proper etiquette in various social situations, but the anxiety comes first. At least, imo. Like you say, in non-stressful situations, you generally don't have any problem. If you hadn't been anxious, you would have picked up those skills; and even if you hadn't, you wouldn't care (like my sisters).
In science, ideology tends to corrupt; absolute ideology [corrupts] absolutely" - Robert Nisbet