I shoot pretty much the same subjects you do (kids/family, landscapes, portraits and some kids sports) and find the 6D to handles everything except the most demanding action photography pretty well. I still have a 7D, but I very seldom use it anymore, just occasionally for outdoor sports or as a backup body for more reach with my telephoto lenses.
I have never used a 60D, so can't compare the 6D AF to the 6D. My first DSLR was a T2i (550D) and later I purchased a 7D then a 6D last December. If find the 6D AF to be good enough for pretty much all my uses. I even started using it for indoor sports (junior high basketball and wrestling) instead of my 7D. The 7D could nail a higher percentage of in-focus shots, but the 6D IQ was much better at the high ISO's required indoors at fast shutter speeds. I'd guess I could get maybe 60% in-focus shots with the 7D at basketball games and maybe 40% with the 6D. For me 4.5 fps is plenty, but I use one-shot AF 95% of the time.
The 5D3 is definitely a better camera, but I could not justify the additional expense on what is just a hobby. For me IQ was my #1 priority and in that regard the 6D is the 5D3's equal. I also find the WiFi feature handy for transferring pictures to my iPad or phone or using one of those devices to remotely control the camera on occasion.