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Being a woman, or a queer person, or ethnicity are not belief systems, they are biological traits.

There is many bigots who claim the contrary and there is an entire culture war going on right now on the claim that whether people being openly queer or women doing "manly" things are doing so from an agenda. Making the same claims against religious people or "deeply religious people" is equally wrong. Also "deeply religious" is an arbitrary concept similar to saying someone is "too queer" or that a women would be "too much out of line" or similar nonsense.

Even financial status is not a belief system.

There is a lot of empirical results that show a strong correlation between wealth and ideologies. Also just very basically, judges for the most part come from higher financial classes which limits the life experience they have to understand the situation of poor people. Again the point is to move from the generalization to look at the specific person in question. E.g. a judge that has a track record of decisions favoring wealthy people probably has a problematic bias, but there is judges who do not show such biases.

But politics can be studied historically and analised socially and economically, and can be chosen based on data and evidence.

Economists literally have "schools of thought" that they choose to follow. Those are believe systems. Any good sociologist, ethnologist, historian, anthropologist and other humanities researcher will acknowledge that they are biased and that their analysis and interpretation remain subjective. It is very common to have two diametrically opposed interpretations of the same set of "data and evidence".

There are thousands of religions in the world, each one believing different things, each one claiming to be The Ultimate Truth and - what’s worse- to be above human laws. Picking one is completely arbitrary (usually determined by upbringing).

That is a very generalized claim that seems to come from a place of anti-religious belief, rather than an empirical analysis. Very basically there is more than 4 billion religious people in the world. If a relevant number of them would believe to be above human laws, society would look very different.

Judges working under the influence of different religions rule differently on the same topics.

So do judges of different gender, ethnicity and other group identities. If you believe that these play no role, then you would see no need for representation in higher courts.

I’ll prove you that qualifications alone are not enough: if I had the appropriate qualifications to become a Supreme Court judge, but I claimed that the Sacred Pink Unicorn,

You are arguing against a straw man here. I said: "we should look by their qualifications and act if they show clear biases making them unfit"
As you make up the straw man as a generalization against religious people as a whole, or your arbitrary category of "deeply religious people", you are engaging in bigotry as you generalize onto a broad group of people. The same can be seen by the culture warriors attacking women, queers, racialized groups...

All of your arguments are subjective and the result of your prejudice against religious people. I think these prejudice are working by the same mechanisms like prejudice against other group identities and i hope that this helps you to question and overcome them.