My understanding is that both Civgov and Milgov see themselves as the rightful, legal and only real government of the USA. Therefore, any move away from even symbols of that would be avoided at all costs.
Without the symbols and trappings of the prewar government they are little more than yet another bunch of upstarts trying to forge their own empire. While they use the flag, have the constitution in their posession, declaration of independance, and/or dozens of other items and locations of historical and political significance, they will be viewed by many as legitimate.
The moment they mess with that, they become clowns.
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