FOR THE MOST PART, I believe Americans - indeed, all non-Israelis and Israelis "hu"l" - should stay out of Israeli politics.
Let there be no mistake, the Israeli rabbinute IS political, with all the problems inherent in anything political.
BUT, perhaps a suggestion based on what seems to work in the U.S. and other western countries.
LET CONGREGATIONS SELECT AND PAY FOR A RABBI OF THEIR CHOOSING.
Riskin and the Jerusalem-based Chief Rabbinute are at odds because, from what I've read of Riskin's practices, he is too liberal for the rabbi/politicians of the Chief Rabbinute.
Riskin, for example, is "in bed" with the Tzohar Institute; this organization develops technology that allows observant Jews to live in modern times within halakah (Jewish religious law). As an example, it found a sound amplification system that does NOT require electricity that can be used on Shabbatot and hagim. Riskin also founded a bet midrash for - gasp - women where they will be educated to near rabbinical level without being rabbis. A fuller list of Riskin's "sins" is given in Isi Leibler's blog entry titled "Dissolve the Chief Rabbinute now" that I commend for your education. (One Riskin offense I read about some years ago was to include the mother's name when a man is called to Torah - So-n-so ben (Father's name) and (Mother's name). It seems to my non-rabbinical mind appropriate since the child's first and perhaps greatest influence, certainly during the child's formative years, is the child's mother; not the father and not a teacher; the mother.
According to Leibler's post, the city government wants to keep Riskin as its chief rabbi, and Riskin is willing to serve sans remuneration. He's 75 years old and surely has a state pension so he can afford to serve gratis.
THE PROBLEM in Israel is that rabbis are paid by the state; everyone pays taxes to support the rabbinute, even helonim, even agnostics and atheists.
If the rabbinute loses its ability to pay its rabbis - its employees - salaries will have to be paid by the congregations the rabbis serve. If the community wants a different religious leader - a Shammai instead of a Hillel or vice versa - they simply can hire one.
There is, of course, another side to the coin; a few wealthy people might be able to control the rabbi, but then a few people control the paid-by-the-taxpayer rabbis now; the difference is that now the power rests with the Chief Rabbinute in Jerusalem, not with the people in, say, Efrat or Holon or Yavne or Bet Shean.
Even today people "vote with their feet" by simply going to a different minyan. (In Bet Shean, for example, there are at least 4 minyans within easy walking distance of my Mother-In-Law's home. Where I live in Florida there also are 4 such minyans, albeit only two are Sefardi. With the possible exception of the Aish minyan, all rabbis serve at the pleasure of their congregation.)
The "American way" hardly is perfect, and certainly there are politics involved, but here the politics are local while in Israel the politics are isolated to a handful of selected rabbis of like minds; incestual.
Since the death of R. Sholom Rivkin in 2012, St. Louis - the last community to have a chief rabbi - the office does not exist in the U.S. According to the rabbi's obituary, the late R. Rivkin was chief rabbi from in 1983; he retired in 2005.
Apparently as with Efrat's chief rabbi, St. Louis' chief rabbi also was in the forefront of modern orthodoxy. According to the obituary in the St. Louis Post-Dispatch, Rivkin often broke new Jewish legal ground. In the 1980s his decision to allow a St. Louis Jewish woman to undergo in vitro fertilization influenced Jewish legal thought on bioethics. .
Bottom line: Let congregations name, and compensate, the rabbi of their choosing. It isn't a perfect solution, but I suggest it's better than what Israel has now.
For a "taste" of R. Riskin watch the Balak video on YouTube.
