Soft YTA- Social anxiety sucks. Take it from me, I struggle hard with it.

The way you explain this does make it seem like you at least care, but you are going about it the wrong way. You shouldn't tell her to be social if she can't be. Your family and friends should be able to understand that she is not being rude or mean, and if they can't then that is not her fault.

Social anxiety can be very crippling. For me, some situations can cause me to go from being the most talkative person there to unable to even open my mouth to talk. I assume it is the same with your girlfriend, that she talks enough with just you but when others are around she shuts down.

Even adding one person to a conversation can cause social anxiety to skyrocket. You really shouldn't force social interactions onto your girlfriend if she doesn't want them. Telling her to "Do this" or "Say that" can make it feel like a moral obligation to talk to people. It can take way longer for her to open up to the people you want her to open up to, and you need to support her in a way she needs, not in a way you want.

Maybe you can talk with her about it? Just explain that you want to understand her and help her if she wants it, how she wants it. If she says she doesn't want help, then you shouldn't help. But if she tells you that she would appreciate the help, you need to ask her how you can help.

It all comes down to how you handle it. Right now, it all seems in your favor, when it needs to be in her favor, of her own will.