I doubt that's true, in fact, could be the reverse. SS is not enough in many cases for poor people to live on because they have less in savings, but if they earn too much working, they can't qualify for SS. They also have shorter lifespans. Middle class people in general live longer and can afford to retire earlier because SS complements their savings, which transfers to them the taxes on the wages of the poor.
The gap is large enough it's really just math. TLDR; Some people get 70+% more per dollar put into the system.
Further, there is no income cap on social security, you simply get deferred payments, but at 70 earnings stop counting. AKA if you take it at 62 and make enough money to get 1/2 as much from SS at 70 it's recalculated and you get more money than if you had started at 62.
Double earnings to 80k and keep other number the same and you get: $2,510.00 which is much less than 3,128 or 2x $1,564.00/month. Drop that to 20k and it's still $1,090.00.per month which 73.7% better ROI than someone making 80k per year.
Now yes lifespan is critical, but not enough to make up the 73+% gap. Marriage on the other hand can provide more than a 50% benefit on it's own which shifts things even more dramatically.