IIn my arrogant opinion your argument is arse about face. You don't make things fairer for all by removing help from those who need it, even if that does put both the young and old in an position of equal disadvantage.
Most of the time that you're working age, you pay NI or are credited with it (eg for signing on, or meeting the requirements for Carers Allowance or Income Support if a single parent). Shared contributions over a lifetime should result in benefits whenever the situation of extra need arises.
Are children undeserving because they haven't had the chance to pay NI? What about school leavers and those who've recently left further or higher education?
A short clear form which boils down to "Do you need this? Sign here." will get a far bigger uptake than a 50 page form asking for details of any relationship, any children, your housing, your savings, and lastly (I found this one on the "fix yourself a break scheme for carers" form", which I failed to complete) up to 500 words on the difference that getting this payment etc will make to your life.