I think (as a dad for almost seventeen years and a semi-serious hobbyist photographer for fourteen years):
* The A6300 is a great choice. However, unless cost doesn't matter and/or you really want the absolute best video quality, the A6000 would be 85% as good for 55% of the price ($548 versus $998 for the body--but see the next point).
* Definitely spend the extra $150 to get the E 16-50mm f/3.5-5.6 OSS kit lens. If nothing else, it will give you a very compact, versatile zoom lens for when you want to reduce the bulk and/or weight you're carrying--definitely an issue when also bringing a baby along!
* The Sony E 16-70mm lens is really not that much more versatile than the kit lens. The long end of 70mm is more useful than 50mm, but not radically so. It is only one stop 'faster' (letting in more light) at the long ('zoomed-in') end, and is actually 'slower' at the wide end. It certainly does not appear to be anywhere near good enough and/or enough better than the kit lens to justify its price ($998). But if cost doesn't matter, sure, buy it in addition to the kit lens (it is significantly bigger).
* I would want a lens for lower light levels and/or more blurred backgrounds than either zoom can give you. I agree with another poster that babies are small and a longer lens is better. The Sony E 50mm f/1.8 OSS would be a great choice at $298. But if you really want a more 'normal' lens, then the Sony E 35mm f/1.8 OSS for $448 would also be a good choice.
* You may feel like you're clueless, but don't be afraid to experiment. And take a ton of shots during the first year--babies change so fast, and by taking a ton, as they say even a blind pig finds an acorn once in a while. If you take 100 pictures and get 3 real keepers, be happy, you have 3 keepers of your infant son.