Unless I'm misunderstanding you, that first link shows that, for people who get pensions, the pension is in the same ballpark as Social Security. (Private pensions tend to be on the lower side but they're still close to $1,000 month which is about 2/3 of the Social Security payment in that table.)
1. http://www.pensionrights.org/publications/statistic/income-p... 2. https://www.cbpp.org/research/social-security/social-securit...