We acquired two puppies when we had primary-aged children (we got the first one when two of the DC were pre-primary; the time we got the second, everyone was at school).
I won't lie, it was bloody hard work, especially the first puppy. I worked P/T from home, DH was out of the house 12-13 hours a day, and we had no family nearby to call if we needed to go out, or to mind the DC if one was ill and couldn't be left, but the dog(s) still needed walking. Both dogs got plenty of exercise, and though both of them were fabulous with people (including small children) neither was particularly well-trained. Finding the time and the headspace for training was the hardest thing of all.
On the plus side, both of us had grown up with dogs so we had a very clear idea of what we were letting ourselves in for. Two walks a day were non-negotiable, which saw me many times out in the winter dark through snow, sleet, horizontal rain etc.
But it worked. Two dogs had happy lives and our DC grew up into massive dog-lovers. None of the DC ever got bitten, or even came close. You have to keep an eye, teach the DC to respect the dog's space and when they're eating, encourage plenty of positive interactions between the puppy and the DC.
You do have to go into it prepared to build good relationships between the DC and the puppy, and explain to the DC that toys and other possessions left lying around might get chewed and if they do, it's not the puppy's fault. And you have to be sure that you WILL give the puppy enough time, exercise, play etc. We were/are by no means unique in successfully combining young DC and a dog. I only know of one family who gave up on a dog they acquired when they had young DC, and that was an enormous young rescue dog who should never have been rehomed to a family where both the adults had compromised mobility...
As for breed, I think border terriers are a good shout. Just be prepared for a LOT of hard work and some frantic shuffling of schedules to make sure that everyone, including the dog, gets what they need.