In answer directly to your OP, toggling the state pension age to fill the gaps in social care would be incredibly poor value for money for the tax payer.
The cold reality is that other than a recent small blip (due to covid and lockdown harms) the general trend for life expectancy is on the up. A more accurate measure of the problem would be life expectancy from age 65. Once you are at or around state pension age, life expectancy is 18-20 years so the from birth figures are largely irrelevant here.
We are now increasing state pensions year on year by a large amount due to inflation, and there are more and more people in receipt with less and less in proportion paying in. Without further tax increases and it is just not sustainable.
It is a very difficult sell here because there are lots of conflicting expenses for people, particularly but not exclusively for young people, but if you don't want to be forced to work until state pension age you will need to save to provide for yourself. For some people this will be easier than others, those in the NHS/teaching etc. have very generous schemes where benefits can be taken (rightly at a reduced rate) long before state retirement age, however others may need to make greater sacrifices. I know it feels unfair, and those who have made plans for certain ages will rightly feel aggrieved but given the circumstances we find ourselves in I see little option but to act.
My old dad for example had a humble job and we always seemed poorer than guys who you suspected had lower salaries, my builder dad friends had new cars and wore designer gear. But my dads sacrifices allowed him to retire at 58 while the builder dads are still grafting and blaming the government for how unfair it all is.