You, are an awesome mom. As I read through your posts thinking of things and suggestions, you had already covered most of the bases.
Girls at this age, in my opinion, can be gangs, when it is more than two. To your daughter, maybe she is just the type of kid who does well one on one, but to add a third child (or more) leaves her floundering, insecure and feeling threatened. She does well one on one I take it, and that doesn't mean she lacks any social graces, or that there is something wrong with her.
Shyness and insecurity may be fueling this. She is comfortable playing with one child, but stepping outside that comfort zone, when others are added. It changes the nature of the activity, the interraction, the competition. Kids at this age have a new best friend every other day, and when you suddenly feel comfortable in your own skin with a person, somebody else jumps into the mix, and it changes everything.
I would respect her development and comfort levels, for what they are. I would continue to discipline for rude behaviour, but try not to push too hard to have her feeling there is something wrong with her, because she is not like the other kids in the neighbourhood. Maybe her maturity level (as you've described how she is with adults) is beyond that of her friends, and part of her frustration is just that fact alone.
Her frustration level may have nothing to do with being hurtful or mean to other chidren, maybe it is just more that, at age 7, she isn't getting through to people that she'd just rather entertain herself, or have one friend to play with at a time. Thus, a typical 7 year old response comes out as anger, because she is not mature enough to entirely express herself.
I have found that parents do the group think thing. Goals and rules of behaviour are set for the kids, instead of letting them find their own way, or decide or show, what theyy are more comfortable with. I've seen too many parents push kids into social and sporting situations when clearly they don't want to. As though meeting these social markers, whether welcomed or not, are expectations that have to be met, and met with success.
If you think of the adults you know, probably most of them are not social butterflies, comfortable in large groups, or enjoy being obligated to engage in socializing. We accept this in our friends as being preference, or choosing what makes them comfortable, and likely enjoy their company one on one, but for some reason, we expect all children to meet those social markers, in order to be well adjusted adults.
Your daughter will likely meet, eventually, a more compatible mate, and prefer the company of a few good friends, rather than a bunch of kids who are put together, just for the simple reason that they should all play well, get along, and like each other. I think that is unrealistic.