I’m definitely on the younger end for a mother of a 12 year old since I had him extremely early, so I feel like I’d be less embarrassing than other moms.
I kinda don't really understand this angle.
If you're 12, I would've thought that if your mum is 30 or 60 they're still an "embarrassing" parent.
Your kid wanting their own space with their friends isn't a reflection on your relationship with them, it's just their own growth.
Based on what little information we have, my guess would be that at some point your son has been to someone else's house and their mother stayed in her room and they talked about how cool that was.
I think the only answer here is to explain that you were looking forward to meeting his friends, but you'll do your best to give them some space, but you're not going to be confined to your room.