The socialist subverts society - subverts freedom - by equating "society" with government. Even though we have freedom, we are not free to do anything; even though we are required to do some things - e.g., pay taxes - we are not required to do everything which is not illegal.Freedom is the difference between what is mandatory and what is illegal; in a perfect tyranny that difference would be zero. Which would be another way of saying that there would be no difference between "society" and government.
If there is a difference between society and government it must mean that there is an authority above the government. And expressing allegiance to the government "under God" is expressing allegiance to the premise that government may not presume to be all of society.
There is no difference between that logic and the logic which drove the signers of the Declaration of Independence to appeal to "endowed by their Creator with certain inalienable rights" and other transcendent references. You cannot rationalize rights not granted by government without appeal to authority other than government.
The Soviets had a constitution which purported to be better than the U.S. Constitution. But the Soviets also had no respect for any authority above the state, so their "wonderful" constitution was not worth the paper it was written on. And it was to express that fundamental difference between the Soviet system as it actually functioned and the U.S. system that the Eisenhower Administration advocated the change which you denigrate.
If there is a difference between society and government it must mean that there is an authority above the government. And expressing allegiance to the government "under God" is expressing allegiance to the premise that government may not presume to be all of society.The problem with adding "under God" to a patriotic affirmation is it affirms a belief in a single Deity and the premise that the Nation is subordinate to that Deity. Neither sentiment exists in the Constitution or anywhere in US law. It is a religious statement that has no place in an affirmation of loyalty to a nation that promises and protects complete religious liberty.
Your point is that the government may not claim authority over all things. I'd agree of course, but this statement does not make that point. A totalitarian Islamic regime, for example, could also claim to be "under God".
-Eric