How? Because they can. It keeps costs down and the NHS can't [0] refuse to treat somebody, even if they're coming from a private surgical ward.
I'm not saying that's what's happening here, but this is one reason why private elective care is so cheap in the UK. The other being that they have to compete with £0.
An @nhs.net email doesn't mean "this person/organisation is a part of the NHS", it means they provide services to the NHS and need to deal with patient data.
Per capita spending seems like a misleading basis for comparison to me. US government healthcare schemes cover about 25% of the population, whereas in the UK the NHS covers essentially 100% of the population.
The US tax payer already pays as much or more as the UK one, all the extra payments are just a "bonus".
Sure, it's not 'free'. But when you pay taxes, everyone doesn't pay equal amounts. Those who are able to earn more money, pay more taxes. Those who can't pay less. But no one is left to fend for themselves. Even the poor can get treatment. We're also not paying anything like the insane prices you pay in America.