You're right, but it isn't a helpful comment for the discussion at hand. The story in that article, and many like it, are symptoms of a larger problem, but it's not because of public vs private healthcare.

The system works, assuming it's adequately funded, which it hasn't been for a while. Salaries aren't high enough to live on (thanks to the conservative government and creeping privatisation), and then brexit and other immigration policies have caused a shortage of trained professionals and no one willing to fill the vacancies.

There's a whole heap of issues (I've left out quite a few) stacked up against the NHS, and yet it powers on, doing its best.