Rav Yosef in his shiur last night criticized MK Itamar Ben Gvir (Hatzionut Hadatit) for going up to Har Habayit.

Perhaps most egregiously Rav Yosef called him an idiot. I think it behooves a respectable position like the Chief Rabbi, any rabbi really but especially the Chief Rabbi, to speak with respect and dignity, even with those with which he disagrees. I know it is nothing new, but I truly wish he would not sink to the level of name calling and the like.

Besides for that, Ben Gvir has been going up to Har Habayit for a long time, and he has been causing provocations (on Har Habayit and elsewhere) for a long time. All of the sudden Rav Yosef needed to speak out against him?

With the nation on the brink of dispersing for elections, whether the collapse will be this week, next week or next month, it seems clear Shas is feeling the pressure of Ben Gvir's party skyrocketing on the polls with many Shas [and UTJ] voters moving to support Ben Gvir due to his more extreme and provocative approach. It seems likely to me that at least one of his goals in speaking out in this fashion at this time is to stem the tide of supporters moving to Ben Gvir's camp. 

One might say dont involve the rabbi in politics, but when the rabbi involves himself in politics there is no reason to not consider him involved.

And yes, I am sure the halachic aspect alone was also a concern of Rav Yosef's. I do not accuse him of being purely political (though I dont see why that is necessarily a bad thing, considering the position is basically a political appointment with the politicians deciding who the Chief Rabbi will be and making deals for support), but it is a strong motivation.