There are ways to compromise here. The Religious Zionists created both the Hesder system to combine military service and learning — some of the greatest rabbis alive today teach in Hesder yeshivot. More recently they created the pre-army prep schools for young people who don’t want the Hesder program. Sherut Leumi is full of frum people and I don’t know why it would be inappropriate for charedim; indeed many charedim already do very similar service in organizations like the various hatzolah organizations and Zaka. I’d read reports of Nachal Charedi soldiers making siyumim. We all know that in no educational setting is everybody really learning; nobody should kid themselves that everyone in charedi yeshivot in Israel are.
Unfortunately, neither side seems much interested in moderation right now, much less compromise.