Just before the Lewinsky soap opera, Bill Clinton made a deal with Newt Gingrich to essentially launch the same Social Security privitization plan Bush proposed in 2005. The backlash from elderly WWII Democrats would have been enormous, and Gore likely would have done far worse than he did OTL.
To be fair, whether it would've passed or not is a whole other matter, considering Social Security is a lot more popular than welfare, & Clinton probably would've faced a massive internal party revolt, no matter how much he may have tried to label it "saving" Social Security.
Had it succeeded, though, it's correct that privatizing Social Security would've likely been devastating to the Democrats come 2000. There would've been some massive political infighting between the liberal Democrats & the centrists who were Clinton's allies. "Clinton sold us out!" would be a pretty nasty rallying cry to run on in the Democratic primaries (& potentially on the convention floor). It might've well be enough to knock out anyone too closely tied to Bill (such as Gore) too. At the very least, it likely would've significantly softened Gore's appeal come November.