No, the money paid in NI doesn't go into a separate pot. It is just put in the general government bank account together with every other source of tax.
State pensions are paid from revenue, not from money set aside for that purpose, so whether people pay separately into a NI fund is entirely irrelevant.
What bothers me is the way government play fast and loose with NI. I think it should be a hypothocated tax and, over the years it should be built up so that, may be in 50 or 60 years, there is a capital invested fund that produces enough income to pay state pension, that uses revenue taxes as a fall back in bad investment return years, but little less.
Birthrates are declining. The number of people in the workforce - unless we allow massive immigration - will fall as well, while the number of older people grows and this imbalance in the age distribution may mean that pensions will have to shrink in value because younger people will be unwilling to cope with ever rising taxation to pay pensions.
Do we wish to impose an impoverished old age on our children and grandchildren, where they will look back and think we lived in clover?