This problem has existed, I believe, since the Israelis formed their government in 1948, if not immediately after they first successfully defended their new state.
It does seem quite a paradox that those who most righteously claim Israel's right to exist, claim the right TO NOT DIE IN DEFENSE OF THAT BELIEF.
Maybe those of us who are not Israeli are missing some of the finer points of this issue.
I think you're somewhat mistaken. The Haredim (Ultra-Orthodox Jews) who dodge the draft and don't serve in the army overwhelmingly come from non-Zionist or anti-Zionist sects. These sects believe that Israel is not a "true" Jewish state because it is not run under Jewish law and/or that it was actually sinful to create a Jewish state without the Messiah. Because they attach no special signifiance to Israel, or disdain it, they dodge the draft and cause all sorts of other nuisances. These sects also reject the most of the modern world, including secular education and technology.
The Israeli right-wing (i.e. those who "righteously claim Israel's right to exist" as you put it) largely consists of Zionist Orthodox Jews, almost all of whom serve in the army -- and, statistically speaking, volunteer in very high percentages for combat units.
The Zionist Orthodox Jews are just as Orthodox in abiding by Jewish law, but follow Rabbinic opinions that justify Israel's existence as a secular Jewish state and believe that, if anything, modern Israel is paving the way for the Messiah. These Jews also very much live in the modern world -- they go to university, get advanced degrees, use the internet, etc.