I was surprised how sudden the change was - I didn't become a dad to my kid, I became a dad to all kids within sight or sound. Sure, most of the time I don't have to do anything, but I certainly notice much more now.

I already feel like this and I don't have kids or want kids.

This. Also, the feelings have become stronger as I age.

So much stronger. I went from completely understanding the "we should have restaurants/theatres/etc. where no kids are allowed" people to finding the notion totally incompatible with a healthy society. I love kids. I want to see them thrive and grow. I want to invest my taxes into their success. I vote for affordable daycare. I vote for expanding school programs. I take kids in my complex to look at stuff with the telescope. I 3d print things with them. I ask them about their lives and wonder what they'll be like. I have so much more patience for them. They matter more than I do, now. That's about it.