I doubt that pension companies assume 82 as the average lifespan. The standard calculation is something like total pension pot divided by 25 (for 25 years of pension).
What most people don’t realise is that 82 is an average, including all those who have succombed to cancer in their 60s and 70s (which pulls that average down).
The key statistic is the ‘mode’, which is sort of the ‘most popular’ age to pass away. For men, modal age of death is something like 87 and for women it’s something like 93. According to the office of national statistics, most recent week available: 2.7k deaths for the over 90s (the most popular age range to die within), 2.2k deaths in the 85-90 age range, and 2k deaths in the 80-85 age range.
As for the statistic (someone quoted 2.5% ?) that only a low proportion of elderly people go into a care home, this is misinformation. The true picture emerges when you compare place of death using the very same office for national statistics website. For the most recent week, roughly: 3k people died at home, 5k in a hospital, 0.5k in a hospice and 2.6k in a care home. That’s about a third of all people dying in a care home.
As for the claim that most people only spend a few weeks in a care home, again, misinformation. LSE research says the median stay is 19.6 months.
What the above all means is that, yes, we should plan for what happens when we reach old age rather than assume something will turn up or someone will sort it out for us,