Everything about politics has to have enough footnote and caveats to make people's eyes bleed... but boiling an act down to a single person is not appropriate unless they were the champion of it, or it's intended as a placeholder for the party.
It sounded odd to me that Clinton/Obama would champion this... It felt very free-market / Republican thing to do, especially with it being called the "Privatization Act"... So I looked for more info.
Sure enough, it was introduced by Republicans... But Democrats share full blame on this if you think it's wrong... and the measure passed by so much it's almost pointless to call out the president because -- as you noted -- if the president fought it, there probably wouldn't be unanimous consent followed by 98% approval on amendment. So again: Why call out the presidents and not the party numbers except to inflame?