I agree that accepting and owning it can be a good option, but also I realize rather than fixing the problem it cuold maybe further dig myself into that "I'm anxious, there's something wrong with me, etc." rut and I'll continue to struggle initiating social interaction and strengthening connections. And that's not who I want to be, I want to be seen as friendly, outgoing, and approachable.
Certainly every person, and every person's social anxiety is different, but it definitely did the opposite for me. It made it easier (though still not easy) to initiate or participate in interactions.
It is also not who I'd want to be if I was crafting myself from scratch, but we don't get that option. It is who I am. I'm stuck with it, just as I'm stuck with being 5'4" even though I'd love another 3-4". That'd sort of the point. Wishing to be different implies that this is a flaw or a brokeness, and it's not. And working accepting that for myself, helped me accept that point.
If I say that I'm anxious around new people, it's not interpreted as being not "friendly or approachable". Actually, I think it makes me more approachable, because people see right away I'm not putting on airs. That makes me more approachable, not less. And I think it also shows a certain self-confidence, to be able to admit right away that I'm not good at something, and to not be ashamed or embarassed by that fact. People generally seem to respect it, and respond quite well to it. And then if/when I'm stuttery or weird or awkward, they know it likely isn't about them or my opinion of them; it's about me being in a situation where I'm not at my best, because I've told them that.
I used to want to not be socially anxious. To aspire not to me. "Gosh, this is so easy for so many people. It shouldn't be hard. How do I make it not hard? How do I make myself enjoy new people and want to socialize and not be so uptight and in my head in these situations? What is wrong with me, and how can I fix and change it?" That's why I went to therapy. But what I learned is that this is almost as hard-wired as me being 5'4". I'm not going to significantly change the fact. But I can make it far more comfortable to be socially anxious, and I can make those around me when I am socially anxious far more understanding and comfortable themselves. Which, in and of itself, relieves some of the anxiety, almost magically. It's not the shameful secret struggle anymore when I name it out loud. For example, maybe I stammer at an intro, or say, "you too!" after they ask me if I'm enjoying the party, or whatever. And then I say, "Gosh, I'm sorry. I'm so bad at this. people-ing is not my super power, for sure!" and everyone is at ease, including, to some extent, me. And I'm seen as approachable and friendly, and maybe even kind of outgoing, which is exactly how you said you want to be seen.
If think if you focus on who you wish you were, rather than who you are, you actually feed into the cycle of shame and discomfort, and it makes the anxiety worse. Or at least it did for me. YMMV.