I think in theory national insurance was meant to pay for state pensions when it was introduced. I seem to recall reading that it doesn't cover it now but someone with greater knowledge than I on that front could maybe confirm.
Yes, Employee's National Insurance is a separate unrelated calculation to income tax so the income from which it comes is effectively taxed. Of course, National Insurance is not limited to Employee's and the Employer's element of it (which is bigger) has certainly not been taxed.
Notwithstanding whether it is or isn't a fund derived from taxed money, it still becomes income to the taxpayer and I still don't see why an arbitrary age should bestow a tax free status if it's large enough to otherwise be taxable? People are also taxed on interest earned on savings that come about from income that was taxed at source when they received it. What's the difference between that and a pension if it relates to whether or not the funds were once taxed?
Even if the answer to that is nothing, and therefore by logical extension, that savings interest shouldn't be taxed either, then it would just mean higher income tax would be needed to meet the state tax burden.
I'm a chartered accountant incidentally, though an auditor more than a taxation expert. Most of my tax experience is in corporate tax or payroll stuff.