This is a weird article that starts out with a customer service problem and ends up extrapolating an entire new health system out of it. The problem this person has is that they signed up for an HMO instead of a PPO, which is what most people in the US have. HMOs subject you to additional red tape in exchange for lower premiums; PPOs allow you to walk in to a specialist (though most Americans will see their GP first anyways).

Whether you get an HMO or a PPO, it is far easier to see a specialist in the US than it is in the UK. In fact, most health care economists would argue that it is in fact too easy to get specialist care in the US, and we as a result overconsume services without obtaining better outcomes.