The OP asked about “valid. logical” reasons.
Almost everybody in this thread is right, of course, about two things.
1.There is no valid reason for exempting the religious fanatics from the same civic duty that other citizens face.
and 2. the actual reason is realpolitik.
But there is a “logical” explanation. And to understand that, you need to know the context of how the ultra-religious live. So for those who are as the OP says “non-Jewish, non-Israeli, and only have the most superficial understanding”, here is my contribution to the thread, for context:
The Haredim (Hebrew for “pious”) live a lifestyle that can only be described as a fanatic cult. It it an absolute, all encompassing lifestyle which completely controls every minute of your day, and every day of your life, and every stage of your life. Like most cults, it requires that you have virtually no contact with regular society, and punishes you and your family if you stray from their path..
Their lifestyle is, of course, based on the rituals of standard Orthodox Judaism. Many of these rituals are common knowledge. If you have met any Orthodox Jews in America, you may be aware of them: wearing a yarmulka on your head, eating only kosher food reciting prayers three times a day, strict observance of the Sabbath day of rest.
All these can be done while living a normal life.
But the Haredim take it to extremes. They deliberately cut themselves off from the wider society around them. They honestly believe that working a job is unnecessary, because studying the Bible and the Talmud* 10 hours a day 365 days a year is doing God’s will.
To illustrate how extreme their lifestyle is, I’ll discuss their attitude to marriage.
A huge part of the Haredi lifestyle involves family life. Having 5 children is normal, 8 is common. Raising all those children, ensuring that they all live the complete lifestyle is a huge part of the cult mentality. Every child is raised knowing that he or she will get married and have another 5 or more children.
And this is one critical aspect of their society which is often ignored by secular political commentators.
Getting married is THE single most important goal of Haredi cult society,
Many, perhaps most, marriages are arranged by the parents. Every girl at age 18, and every boy at age 20 is expected to get married. Typically, the parents arrange for one or two dates, in which the boy and girl will meet for conversation. They chat, and if they decide that their parents made a reasonable choice and that the two of them seem to be generally compatible, they announce the wedding date a few weeks later. (Love is irrelevant…that comes later, after a couple years of living together and raising the first few children.)
A year later the first child is born, soon followed by many more.
And now (if you are still reading!)—back to the subject of this thread:
One main reason that the Haredi community refuses to serve in the Israeli army is that serving 2 or 3 years would destroy the entire system of marriages, which is the bedrock of the Haredi society.
It would remove the young people from their cult, expose them to the real world, and destroy the mentality which all cults are based on–being locked into the closed world of the cult.
This is a huge challenge for Israeli society. It is not just a question of changing a law or two regarding drafting Hared members , or cutting the funding of their educational institutions. It is an issue of forcing massive social upheaval, breaking the mindset of the cult, and destroying the lifestyle which they have developed , becoming increasingly more and more fanatic over the past 2 generations. (About 40 years ago, the Haredi lifestyle was less fanatic)
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(*The Talmud is a truly vast body of Jewish law, which takes many years to learn casually, and takes a lifetime to know deeply.)