Separation of religion and state

I have been grappling with this question for a long time. On the one hand, logic says that religion should be separated from the state for the benefit of both the state and religion (some say that this separation is one of the main factors in the continuity of religion in the US, more so than in Europe).
On the other hand, it is not clear to me how this concept is possible, not only in Israel but everywhere. Since the principle of separation of religion is relevant to any ideology whatsoever, since no one wants any kind of ideology, not just religious, imposed on them, it turns out that it is impossible to involve any ideology in public policy and institutions, which is clearly not feasible.

(This inherent contradiction can be seen in the American education system, where it is forbidden to teach about Christianity but is required to teach progressive ideas under the guise of gender education and African-American history, even though both are widely disputed in public).

In Israel, there is another problem: the very definition of Judaism for the purposes of the Law of Return is strictly a religious matter. But perhaps this issue alone can be excluded from the general separation.

What do you think?

Indeed, American segregation is absurdly done. But it is possible to teach things without preaching about them, and to bring in the different opinions about them. On a philosophical level, it is not possible to define this clearly. Isn't education for tolerance an ideology?

A. The suggestion to teach things without preaching to them is not practical. Most people have difficulty presenting two opinions equally when they are clearly on the same side, especially on ideologically charged issues.
B. Most people do not agree that their children should be taught different opinions equally, when they have a firm position on the matter. Supporters of science do not agree that they should be taught “unscientific” theories (like the story of creation) not even as an option, and religious people do not agree that they should be taught heretical theories.

Education for tolerance is indeed an ideology, and as such there are many disputes about when and how it should be implemented. We are discussing this too.

Apart from education, the separation of religion and state is relevant to determining public policy. It is impossible to determine policy without a worldview behind it, and there are countless arguments about this. So how exactly do we separate ideology from policy?