I played Forager a lot when my kid was new. I enjoyed the fairly engaging gameplay and the newborn liked all the colors, shapes and sounds.

I also sympathize with the latter half of your post. I was only promised a small amount of paternity leave but, very shortly before my kid was born, my company was purchased by another company that only offered paternity leave to employees who had been there for more than a year. At the time I had been there nine months (I have now been there seven years). I was pissed. Fortunately my direct manager was understanding and gave me a reduced workload for a while.