This is correct. They are employees, employed to play football, their wages for football duties are taxed at source as salary in the same way as most workers in the UK.They are employees. Their income should be subject to tax and NI like any other employee.
They have to wear a uniform, have no control on when/where they work and all their facilities are provided for them. No different to a bin man (person).
That’s why they have a salary paid through PAYE.
When it comes to image rights and other income, that’ll be paid via a company. Then profits paid via dividends when the player wants. Which has different tax rules and no NI paid.
But other work they do outside of their employment contract (which includes image rights) isn't regarded as employment income. Usually (but not exclusively) the image rights sit in a company and the fees are paid to that company. Depending on where it's established, that company will still pay tax (but at a lower rate than the 45% income tax on salaries) and if the company pays a dividend or salary to the player then that will be taxable when paid in the normal way. Usually the players leave the cash in the company and use it to fund business investments (which in turn would pay tax on any profits they make).
Typically, the players aren't avoiding paying any tax at all on their non-employment income, but they do have structures in place that mean they pay at lower rates (or pay tax outside the UK), hence HMRC's interest in their tax affairs.