Stick with Nex or Head Back to Micro Four Thirds

What you are describing is one of the most challenging photographic scenarios imaginable:
1. Fast moving subjects so you need to use a fairly high shutter speed.2. Low light - and you don't even want to use a bounce flash.3. Want to keep ISO low.4. This means you need very fast lenses - which creates its own challenges in itself as depth of field becomes very shallow at wide apertures.5. Which means your focus needs to be spot on - hence your desire for better AF.
It seems to me that all these factors are working against each other and something's got to give. I don't think you could have done much better than the Micro Four Thirds setup you had really, but now you have the NEX, a couple of comments:
1. I don't understand why you don't want to use your bounce flash. I got one myself for just this purpose (taking photos of my kids indoors) and love it. I like using available light too but find indoor lighting at home is seldom well positioned to capture subjects (my kids) favourably.
2. What ISO are you comfortable going up to? I don't hesitate to use ISO 1600 - and will also use ISO 3200 at a push. Using RAW and some decent noise reduction software achieves perfectly good results as far as I'm concerned. If I get too caught up in pixel peeping I just remind myself that it's a 16 megapixel image I'm looking at and the largest print I have ever made is 8x12", which at 300dpi is still only 8 megapixels.
I picked up DXO Optics Pro recently when they had a $99 promotion (it's usually $169) and I'm very happy with the results I get from the RAW files from my NEX-F3. I hear a lot of people talking up Lightroom but I tried it and the results I got weren't anywhere near as good as DXO.
I'm not saying any of this is a substitute for fast lenses, but I have a young family like you and as much as I would love to have it I can't imagine any way to explain to my wife why it's so important for me to get the Sony 35mm f/1.8!
Good luck in your decision making anyway and please let us know what you decide and how it works out.