I wish I could do this...sometimes :/
I don't want to project onto other people here, but I wonder if this is really so hard for us. There are times I've approached people in public, or been able to joke with them, and had no problem. I was also recharged and not in a highly stimulating environment. Even with Allistic people this was easy, in small doses.
But, if we're interacting with those who communicate differently, for long periods of time, this will certainly be taxing. If we're in places which cause sensory overload, this will drain us, too. So it's easy to think we're no good at socializing, or being spontaneous, because we're rarely allowed to try. Instead, we're told we have "social anxiety," which is like this vague, judgmental catchall. I'm not afraid of socializing, I'm afraid of eye contact. And noise. I'm afraid of listening to another Allistic person monologue about their personal life and ideology, mistaking my quietness for low intellect and patience--not the lack of enthusiasm it is. (Talk about ignoring social cues.)
It isn't "social anxiety," but anxiety from masking. I think what we fear is having to look and act just right, to avoid rejection, tantrums, and abuse. This term "social anxiety" that gets thrown at us so much is just self-serving to me. I think we're perfectly social, especially with other Autistics.