The reason for the 25p upon reaching age 80, is because in around 1948, either at the same time or about the same time that the NHS was introduced, Old Age Pension as it was then called had an additional 5 shillings per week (decimalised as 25p in 1971) paid upon reaching the age of 80.
At that time, that 5 shillings a week extra was a substantial boost to the pension, bearing in mind that £4 a week was a good wage, but although the pension has been increased over the years, the 5 shillings extra at 80 was not, it has stayed at the same level. Now that the £4 a week good wage is now around £400 a week due to wage rises including inflation, that 25p really should be about £25 a week, which would be a good addition.
Also, many more people reach the age of 80 now than did then, less smog, better food, the NHS, better medicines and surgery and so on.
I think it is correct that the until recently male state pension age of 65 was set at 65 because around 1905 or so when that age was chosen, it was copied from what Bismarck had used in Germany when he had started state pensions there. However, I think that Bismarck chose 65 because very few men survived to that age, it was a matter of providing a pension to the few that did.
I remember reading a letter in a magazine some years ago from a woman who wrote that successive governments had kept the pension scheme going and she wrote - and it has worked! I think she was 72 and enjoyed going out to ballroom styled tea dances in the afternoons.
It seems to me that the big problem with pension financing is that when they set out the benefits just after the war they based the age that people would live on data of people passing away at that time, but many of those people had lived in different conditions than people born many years later. And conditions have improved enormously since the late 1940s. Though alas not for everybody as there is great poverty for some people in this country.