The NHS is funded over the National Insurance, which you're not including. You also need to account for local taxes (e.g. council tax in the UK).

But the total tax bills are still not that different, as they are marginal rates for the most part and the threshold where they start are different.

The effective rate paid in the UK is still somewhat higher for most earners, but for most people it's not as much as you might think from just looking at the headline rates.

We also pay taxes for Social Security and Medicare, which are 6.2% and 1.45% respectively. Most of us pay for at least part of our health insurance premiums as well.