RSRH: R’ Hirsch was referring to religious institutions in his times which were perpetuating anti-Torah values. However, the State of Israel presently doesn’t perpetuate any anti-Torah values; to the contrary, the State actually doesn’t interfere at all in the way of life of its citizens. The goals of the State are solely to provide security for its citizens, and to help them economically, so they may live secure and prosperous lives.
If you mean to argue that by davening publicly for the welfare of the soldiers and the protection of the State, we are indirectly hashkafically legitimizing the Zionist concept that Jews have a right to Statehood; then I hear your argument. However, one can still argue that since much spiritual good has resulted from its establishment, therefore our indirect endorsement can be perceived as an endorsement of its outcome rather than an endorsement of its actual establishment; much like the Posuk praises Yael for her deed with Sisra which the Gemara categorizes as an “Aveira L’shma”. Also, at this point if the State were dissolved, there will definitely be a matzav of Pikuach Nefesh, so there may very well be an obligation to help with preserving the security of the State and its current institution.
Additionally we cannot forget our personal and primary obligation of hakaros hatov to the State for providing us security for the past half century. Even Moshe rabbeinu who was commanded by Hashem to hit the sea/earth, felt that act would be considered acting against his moral obligation of hakaras hatov; how much more can we say about our moral duty of hakaras hatov to the State for providing us real security and monetary assistance for so many years. Definitely this primary obligation overrides any concerns of committing any aveira by indirectly endorsing the establishment of the State 60 years ago!