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"Again, they didn't say he had to renounce his citizenship. They said that if he retained his US citizenship, he could not use his British passport to enter the US, despite the fact that he is British."

Except that he had a US passport, which he apparently allowed to lapse. If he has (or has had) a US passport, he's American, not British. If he's (by law) American, he needs a US passport to travel outside the country. If he doesn't want to get one, either he can forget entering the US on his British passport, or he can renounce his citizenship. Which will likely trigger his addition to the Terrorist Watch List, these days. The US recognizes dual citizenship, sort of, but does not recommend it, and if you want to maintain it, there are certain things you must do, like maintaining your US citizenship and visiting the US regularly. If you intend to maintain your US citizenship, you are not supposed to do anything that indicates you might intend to renounce it, either. Once upon a time, and not all that long ago, holding elected office or serving in a foreign military resulted in automatic loss of citizenship. This could even happen to those who did not hold dual citizenship, btw. It's still a factor in determining intention to renounce US citizenship.

My take on the article is that he is British and happened to be born in New York.
I did not get the impression he has ever resided in the States, he even states at the beginning of the article though rather oddly he is an American citizen basically b accident of birth. I would safe it is a timing issue. I don't know why he had an American passport, but I would suspect with an American Birth certificate it was easier to get an American passport when he was a baby to go home to jolly old England.
I still don't understand the rule requiring entry under an American passport if you are entitled to another countries passport and choose to use the other passport.