Counterpoint, if your child grows up and isn't "internet native" they will be like people who "don't get computers" over the last decade. It's very limiting.
Spending time on the Internet doesn't make you understand computers. (It does make you understand that you get a hit of happy-brain-juice when you press the Play button.) My wife teaches college courses and laments that her students, fully in the always-online generation, are helpless in Excel, let alone something like a command line.
I grew up without access to computers at all until I was 12. I thought I would break my uncle's computer by moving the mouse.
I've had a career ranging from telecom and supercomputer chips to mobile, desktop, and web dev. I do not think my lack of exposure to the internet at a young age has limited me.
One person in my CS degree (which started in 2008) had his first time using a computer a few weeks before said degree which was interesting. He managed it fine.
