Allowing hareidim (masculine intended, hareidi women are already allowed to work) to work without serving in the army and/or recognizing conscientious objection based on halacha (which is certainly relevant for anti-zionist such as Satmar and Toldos Aharon, and indeed, all groups insisted with the two Eidah Hareidi groups) would be an admission by the zionists that there haskafa may not be 100% in accordance with Jewish tradition – and they aren’t willing to do that. Exempting hareidim on the theory they were too busy learning was a way to avoid confronting the issue, but that is seriously uneconomical and greatly annoys the hilonim.
And to a certain extent, the hilonim are saying that they’ve been told to worship the idol of zionist nationalism, and substitute belief in conquest of Eretz Yisrael for any sort of belief in Ha-Shem, Torah, or even basic morality — and exempting hareidim from worshiping this idol suggests that perhaps the zionist avodah is really an avodah zarah.