Stabilization has improved a bit over the years. For example, I know the original 100-400 canon lens had stabilization good only for about a stop or two, and the new mark II version is good for 3-4 stops of stabilization. But basically, I believe that any newer lenses will have similar stabilization. So for a 60D - all of the STM lenses are pretty new, are terrific, are priced right, and all which have stabilization should all have stabilization about as good as any other new lens.
Also, the wider you shoot, the less stabilization is needed or important. I would recommend against hand holding when making video at a telephoto range. You don't need a tripod, but rest the camera on something, brace yourself as much as you can. If you are shooting at 28mm equivalent, that doesn't matter nearly as much as at 80mm equivalent.
Also, the best camcorders, like the Sony BOSS camcorders, have far superior stabilization to any in lens stabilization. Or you could consider getting a handheld gimbal, but they are pricey and probably cumbersome.
Last note: Yes, your 60d cannot track focus during video. You can try to manually focus if you want. A camcorder is the only reliable way to track focus of subjects. The 70D is pretty good, but probably not as good at a camcorder for focus tracking.