I agree that it's a burden, and it's one that many parents likewise share. It is difficult to be as selective with the shutter as you suggest. Kids don't really sit still, and you have to burst during those precious seconds when everything is just right.
In our case a NAS would make a lot of sense. We've tried Arc (with Glacier/S3) and other storage solutions (Time Machine), but I've had my eye on a Synology unit for quite some time. Also, internal storage as you suggest obviously isn't an option for MacBook Pro users like us.