You can read about the 70D's performance here:
http://m.dpreview.com/reviews/canon-eos-70d/11
And the Nikon D5300 here:
http://m.dpreview.com/reviews/nikon-d5300/10
The reviewers said that both cameras are very responsive, but the Canon has a much higher frame rate for burst shooting. They also note that Nikon has particular lenses which have faster focusing. I assume yours is one, but I'm not a Nikon user so I don't know.
But, let's peel back the onion a bit: You're having performance issues photographing your kids. That sounds like indoor, low-light photography. The autofocus is going to struggle in low light, especially with an f/3.5 zoom, and if you're firing the built- in flash, it's recycle time will be very slow.
A few options:
1) Try again in bright sunlight. This lets you know if the problem was dim lighting or if the camera's just slow.
2) You can try a faster lens, such as a 50mm f/1.8 or even f/1.4
3) If you're using flash indoors, get a good Speedlight flash. You want a model with lots of power, a tilting head, and an autofocus assist beam. Use the tilting head to diffuse the light -- this will slow recycle times, which is why you want the highest powered unit you can get ( faster recycle )
4) You can get the Canon instead. It is a great camera, but I think that if you're having lowlight performance problems, the solution will be more light.
Note that these are just suggestions and I can't be certain of the cause of your particular complaint -- I may be wrong.