Very few people are on fixed incomes, though. Social security has a yearly inflation adjustment. You would have to be on a private pension only to be on a fixed income.
According to the SSA, approx 90% of Americans over 65 collect. https://www.ssa.gov/news/press/factsheets/basicfact-alt.pdf
There are still hardships because it's not a full income replacement, but it's a far cry from 'not all are worry-free rich'.
Edit: The majority of the remaining 10% are covered by state level public employee retirement systems who don't accumulate SSA work credits and private pensions. SSI also exists to provide 75% of the poverty line if you have extremely limited means. Here is a paper by the CRS about senior poverty. https://fas.org/sgp/crs/misc/R45791.pdf