Those employees pay income tax as well as VAT on their purchases and stamp duty and inheritance tax and fuel duty etc etc.
And of they paid those taxes from money paid to them for work paid for by the public sector, none of it contributes additional money to the running of the state. Inheritance tax comes from an inherited, of course, so that does but it comes from the dead person not from the legatees.
How do you feel about somebody who works for a private company that is contracted by the government? Are their wages and taxes paid by the tax payer as well?
I've explained that one already, above. Yes. The source of the money is what counts, not who is doing the work. It counts for company too. If they make a profit on government paid work, they pay taxes on that profit, but those taxes don't add to the total money available to run the state and nor do the taxes on the dividend paid to their shareholders.
Are you suggesting that public sector should be paid tax-free to avoid the administrative burden of collection of their taxes?
It would make things too difficult to administer where people have other income, like interest on savings accounts and second jobs. It's probably cheaper overall to allow teachers and nurses the illusion that the tax on their payslip helps pay for the services they provide. It also gives them the same stake in voting for political parties based on their taxation plans.
For what it's worth, income tax only provides about a quarter of government income. The rest comes from other state revenue streams
Of course. Rates, VAT, corporation tax, capital gains tax ..... For all practical purposes in the context of spending £1,300,000,000 a year, none is provided by the public sector.