Having an injunction on naming a child isn't disownment. It's protecting the privacy of the child. Name it, and now it needs police protection 24/7.
Your characterisation is way off track IMO, but it's a popular viewpoint on Twitter and in the Guardian.
That's rubbish. His public life has nothing to do with his private life. The fact that he is a politician doesn't mean the public has to know about his relationship with his children. His public actions speak loud enough for people to judge his quality (or lack thereof) as an elected official.
I think the cultural gap between American and European on this point will probably never be bridged.
Others have commented on how absolutely ridiculous and insane this is, but it also highlights one reason (among many) that we generally have such shitty politicians. If you believe that running in national elections should make every single aspect of your private life open to public scrutiny, don't be surprised if the types of people willing to run for public office only care about increasing their own power.