Given that you use a 24-70mm I assume that you can get close to the kids.
The 5D3 has difficulty in managing shadow in high ISO, so if your scene involves dark background, high ISO with that camera is not a good idea. This has been improved greatly in newer version.
But I own and use the 5D3 as my secondary and still quite happy with it.
On the other end, moving subject requires a lot of light. You can get away with continuous light (window light, bulb, LED lights) as long as that movement is not "big". But as soon as they begin to run, or jump or even clap their hands flash is the best solution as it has the ability to freeze movement (as many already stated).
Besides, flash is portable, easy to gel and has TTL. These 4 advantages made them popular. Not to mention that you don't have to change the existing light (sometime impossible to change), and you can position the flash almost wherever you want.
In beginning of your flash journey focus on freezing the movement, and later on as you get better, adjust your configuration to incorporate ambient light.
My preference is to put the flash on a small light stand and move the stand wherever I want. If wall/ceiling is white then just bounce it, if not then use a diffuser of some sort (a very portable and work nicely in small space is flashbender).
In the beginning, ignore ambient light and use lower ISO so that flash is mainly your light source, but if you put it far enough it will lighten the whole room. Its advantage is that it freezes every movement. Your ISO depends on the distance, you could start with ISO 400 at 1/160th sec, 2.8
After a while increase your ISO to around 1250, 1/100th sec, 2.8 but you will get more and more ambient light in your photos and then it is time to gel your flash to match the color of that ambient light. Gelling a speedlite is very easy, learn to see the color of the existing light and use corresponding gel.
You will begin to pay attention to the movement as well and switch back to the original method when that movement is big, by lowering your ISO to 400 so that ambient won't contribute much.
Later on, you will add another light as background light to give the scene more interesting look.
Remember though, that if subject is far, 24-70mm won't make it and your flash may not be strong enough, then it is time to use/rent prime lenses and high ISO, like 135 2.0