I am wondering if it's not a reason to be with his wife who just delivered their second child...
No it's not. Tony Marinaro spoke about this on his radio show the other day. Players are allowed to be absent in order to be with their spouses when they give birth. The two matters are entirely unrelated. As Marinaro pointed out it's simply a coincidence and Price would have missed playing time anyway while she gave birth.
But the lower body injury is quite worrisome since Price has had a history of serious knee injuries over the past few years. Every time it happened the team would first announce it wasn't serious and he'd only miss a few games. However as we know the injuries were quite more serious than orginally speculated and Price ended up missing months every time.
After his last serious knee injury (the one when he ended up missing half the season) i pointed out that such a serious knee injury usually means a player will have to undergo surgery. But Price elected to forego surgery and opted for a long layoff instead. I predicted he'd end up regretting that decision and that the knee problem(s) would likely return.
And that's why giving Carey Price an 8-year contract worth $84 million was quite a risk to take for GM Marc Bergevin.