UEF would burn the gate if they could. Colonists' reaction is irrelevant to this strategic point, since the destruction of the node would mute them on Sol's side.
I think it might be possible that the BP team could try to go this path. First, you need to prolongue the war and create enough war dynamics in order to get an opportunity. Second, you need a plan of attack. In my perspective, it will be a battle of wits between those who defend the node and those who attack it. Diversions, stealth mechanisms, surprises, etc. will be tools for both sides.
Now, if the node is destroyed, I think that the war either escalates atrociously, with general Steele feeling suddenly very alone in Sol and deciding to conquer Earth to prepare it for the next built node; or it will wither, with many tevs defecting to Earth. It can also be a combination of both and lots of insanity from tev admirals, commanders, etc. Either way, it's an incredible achievement for the UEF forces, and therefore they should be working to get this done.
It may however prove to be impossible. With that in mind, we can also see the return of shivans... and then all hell breaks loose.