On the one hand I’m saying that I want minimal intervention and regulation, and on the other hand I support a body that would issue exactly those kinds of decisions (in halakhic rulings). How can someone both want and not want that kind of intervention in private life?

***that I want a minimum

You’re not formulating the problem, and not by accident. Because you yourself aren’t managing to formulate it. Are you against legislation altogether? Do you think the Knesset, the police, and the courts should be abolished? This is a question of degree—how much legislation there should be, and in which areas. And on that very point, one can also deliberate with respect to the Sanhedrin.

The mandate of the Sanhedrin would cover the entire way of life, in terms of providing a halakhic response and defining what is forbidden and permitted, whereas the government is required to deal only with the necessary public matters (security, foreign affairs, etc.). On the one hand, a body that intervenes in every layer of life and gives the view of Jewish law on everything (to which all who observe Jewish law would look, and whose influence on citizens’ lives would be direct in the “private” sense), and on the other hand a body that I would almost never interact with.

You’re mixing things up. Jewish law intervenes, not the Sanhedrin. The Sanhedrin has to choose what it wants to intervene in and what it does not. As I understand it, it would intervene only where necessary (for example, when you can’t let everyone do whatever they think is right in Jewish law, and instead need to formulate one binding halakhic ruling. Those are rare cases). In any case, minimization or maximization are two types of policy that could characterize a Sanhedrin as well.