I'm autistic and i almost never remember to initiat the conversation, or feel like it's not appropriate. If any of my friends felt like you do, i won't have any more friends. And i love my friends a lot and can't survive without them. Don't over think it. Each has their own way of socializing. It's about the journey, not its initiation :)
I do, just not as often. I can't control it. I do make sure to let them know i appreciate them contacting me though. As much as possible. Also, i don't understand why the beginning of socializing is more important than the rest of it. Why is me being engaged and interested, continue the conversation, showing appreciation throughout, and making it obvious that I'm enjoying it and would enjoy more in the future, why does that have less weight than a "hi"? I can start a conversation with anyone, wither i like it or not. But keeping up the conversation is what matters to me.
You're right though, it is a teamwork thing. Any relationship requires work from both sides. I present my work by showing appreciation to my friends, and so far it seems like it's working in lowering their anxiety and over thinking :)