I think the problem is pretty clearly stated in your first paragraph.
No, if we met, I wouldn't feel like I knew you forever, and I wouldn't feel that I could tell you anything. That kind of intimacy is developed over a lengthy period of time. It grows, like a strong tree. Perhaps if you start viewing yourself as someone that everyone knows deeply, and view yourself instead as a polite, approachable person, that might help. By approachable, I mean having a friendly smile on your face, thanking someone aloud if they hold a door open for you, appearing pleasant. But not like everyone has known you forever.
Trying to help people see the best in themselves, and trying to help anyone who is hurting, are fine qualities, but they need boundaries. Perhaps you are too much of a "fixer", going too far to help someone whose story you do not know. Now, that's not to say you shouldn't say something nice to the greeter in Walmart whom you encounter weekly ("you always have a great smile! Nice to see you today!"), or telling you cab driver as you tip him generously that you appreciate his service. But don't try to find out everything about everybody, and fix everything.
I have never heard of someone begging to go along on a friend's business trip, and assuming the business would pay. How on earth did you find three such people? What kind of business are you in? Are these women in the same business (perhaps you all are Amway distributors or something)?
Start by trying to view yourself as something other than "if we met you'd feel like you knew me forever and could tell me anything". Something along the lines of "if we met, you'd think I was a happy and pleasant person, and perhaps you might be interested in getting to know me. I wouldn't share everything about myself in the first 10 minutes, nor would I expect you to share everything about yourself. I wouldn't want you to tell me everything. But after I encountered you in line at Starbucks, your day was a little better because I could see you were in a hurry and I let you go ahead of me, and I smiled at you because it seemed like you were very stressed". And leave it at that. Take time to grow real friendships.