I'm going to do everyone a favor and just present a quick summary. On the positive side, I'm happy to see that at least those opposing Rav Stav are now attempting to give some explanation:
Rav Aviner - Those who criticize the Rabbanut are Apikorsim.I think this shouldn't be read literally, but more along the lines that you can't wish to head the Rabbanut, if you are basing your candidacy on how terrible it is.
Rav Ariel Bareli - Rav Stav doesn't accept Rabbinic Authority - proof: he didn't abandon his campaign in favor of Rav Ariel. Rav Bareli also seems to think that Tzhoar as a whole has adopted a method of paskening Halacha that is too independent of the "Talmeidei Hachamim".
Rav Avraham Yeshua Tzuckerman - If I understood correctly, (which I admit I always find it difficult to read the "Kav" Yeshivot writings) the ideological objection is that Rav Stav is too eager to please the community at large. This goes against the natural role of the rabbi as a leader of the community, and not as one that is led by the community. Rav Tzuckerman uses various midrashim (including Korach) to talk of the proper duty of the leader to lead and against the argument that the entire community is equally holy.
Rav Micha Levi - Two arguments. The first is similar to Rav Tzuckerman, that Rav Stav is too eager to please the different communities, and is in practice bending halacha to please them. I think the tone of the argument is slightly softer that Rav Tzuckerman - Rav Stav is even compared to Aharon (as opposed to Korach), but the bottom line is the same. A second argument is that too many local rabbis oppose Rav Stav, which would make it impossible for him to lead them.
Hat tip: @RavTzair