I've heard the reason (elected) Democrats have opposed means-testing Social Security is because they fear the public support for the program will drop significantly once people start viewing it as a welfare program.
The real problem is the lack of public support for broad social welfare policies.
I can also understand why they just wouldn't want to open that can of worms to begin with. I would be afraid to let anyone to the right of Bernie Sanders tinker with the inner workings of Social Security. What would start as a well intentioned effort to curtail social security payments to the wealthy might end with the right taking over the entire process and turning "means testing" into the horrible type of means testing that cripples so many other programs.