The whole world isn't going to listen to the Singularity Institute just because they've got a Friendly AI
'Just' because they've got an FAI? Once you have an FAI (and nobody else has a not-friendly-to-you-AI) you've more or less won already.
We've got to deal with politics eventually.
Apart from being able to protect against any political threat (and so make persuasion optional, not necessary) an FAI could, for example, upgrade Eliezer to have competent political skills.
The politics that MIRI folks would be concerned about are the politics before they win, not after they win.
Work done by Lesswrongians could decrease the workload of such an FAI while providing immediate results. If it takes twenty years for such a thing to be developed, that's twenty years in either direction on the good/bad scale civilization could go. This could be the difference of an entire year that it takes an FAI to implement whatever changes to make society better.
As Multiheaded added, "Personal is Political" stuff like gender relations, etc also may belong here.