Bobert, the filibuster rule(s) are set within the Senate itself and can be changed without consultation of the President or any other branch of government.   The Senate's procedural rules for how and when they can change their own procedural rules are fairly complicated, but generally they change them, if at all, only at the beginning of a new Congress being seated (i.e., every two years). The "nuclear option" referred to changing those rules in the middle of a Congressional term.

As for the Republicans being seen as obstructionist (or not wanting to be seen as such), as long as our media -- who tend to be either asleep at the wheel or really ignorant or in bed with the fat cat corporatists -- don't frame things like the filibuster as that, the public probably won't see it that way. Heck, I've even heard a lot of Democratic spokespeople on the TV machine parroting phrases like "It takes 60 votes in the Senate to get anything passed."   Or (speaking of a filibustered bill), "The Democrats' jobs bill failed to pass in the Senate."   So the public, instead of realizing that bills and nominations are not being allowed to be voted on, are left thinking that the vote was held but didn't pass.    That makes the Democrats seem weak or inept instead of making the filibusterers look like the obstructionists that they have been.