Ignorance ... hence, my question: Is the rabbinate the modern day equivalent of the Sanhedrin?
My guess would be, no, because there is no Hebrew priesthood anymore in Judaism and no temple worship.
The Chief Rabbinate of Israel is not the Sanhedrion. That awaits Mashiach.
The Chief Rabbinate is actually a political body within the Israeli government with its powers determined by political negotiation.
Any authority they carry outside the secular Israeli government is dependent on their personal level of Torah learning, which has varied greatly.
The non-governmental Moetzes Gedolei HaTorah (”Council of [great] Torah Sages”) and the Moetzet Chachmei HaTorah actually carry more religious authority. These bodies are made up the heads of the great traditional religious academies or the heads of Chasidic groups. They do not engage in any kind of “interfaith dialog” of which I am aware.