Your post "The City payroll department treats it like regular income and issues a letter at the end of the year stating the total income earned while injured." does not indicate that the sick pay is tax exempt. It is providing you with a breakdown of regular wages vs sick pay wages.
Typically, paid sick leave is taxable as regular income and is subject to withholding. In most cases, if you receive sick pay from your employer on an "as you get sick" basis, your sick pay isn't distinguishable from your regular wages and is subject to withholding at your normal rate.
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