There is no 'fund' as such; but people born in the 50s will have paid NI for decades by the time they reach 60. Each generation pays the pensions of the one before, so the 'fund' effectively comes in each month as the NI contributions come off the salaries of younger people, just as it did from ours. The fact that there is not an actual pot of money is a red herring, IMO - there never was, yet pensions have been affordable until a political decision to defer them was made.
When there were fewer women in work, the contributions of workers had further to spread, yet it was possible for women to retire at 60. Many of my mother's generation, for instance, retired and got bus passes etc, at 60, without having contributed much NI, having left work when their children were born. They also got things like Child Benefit/family allowance that wasn't means tested, mortgage tax relief and so on, all from the NI contributions of those in work.
I am, of course, generalising, but in subsequent generations more women have contributed, and more will be doing so now. There have been many cuts to universal benefits such as child benefit or MIRAS, so I just don't buy the line that paying pensions is unaffordable.
Also (and changing the subject a bit), as has been said, not everyone will get the full amount of the new pension - it depends on NI contributions - so initiatives to raise the threshold at which people start paying NI should be treated with caution. It might save the low paid a tiny bit of money now, but it will cost them dearly in the long run.