Privatizing Social Security would've likely been devastating to the Democrats come 2000. There'd have been some massive political infighting between the Progressive arm of the Democrats and the elitists who were Clinton's peers. "Clinton sold us out!" would be a pretty nasty rallying cry to sweep the Democratic primaries and Convention hall floor. It might well be enough to knock out anyone too closely tied to Bill - such as Gore. At the least, it might significantly soften Al's appeal come that November.
Gore was Mr. Lockbox in 2000 in regards to Social Security. He probably would not have been the candidate if the White House went for SS reform and he didn't go along with it, as the party brokers that got behind him in 2000 took their orders from Clinton. The Progressive wing of the party was utterly discredited by the defeat of '94 and Hillarycare and all of that, while the Blue Dog and Centrist wing was ascendant because of '96 and the success of neoliberalism in the late '90s.
The Beer track/Wine track split was there, but its tough to see how it would break down in the primary. The Republicans had a fairly tame primary, while the Democrats had a nonexistent one.
Gore to my knowledge did not back social security reform at any point. He was willing to go along with some more centrist measures like NAFTA and welfare reform, and actively backed both of them, but SS reform was a bridge too far, I think.
But Gore was not losing that primary unless Clinton wanted him to.