Regarding the thing about neutral systems, I would point out that such a thing has historical precedent within the past 200 years: the state of Kentucky elected to remain a part of the Union but declared itself neutral in the US Civil War, and its government remained officially neutral until General Polk occupied Columbus.

As to the suggestion that such neutrality is treasonous, that really depends on how power is shared between the galactic and system governments under the Republic - there are certainly real-world governmental models under which it could be the local government's right to decline to participate in the war - and I would further point out that a similar argument can be made regarding conscientious objectors.

Beyond that, depending on how the laws of the Republic are set up, the legality of the war to crush the Separatists may be open to question, especially as the war was to appearances initiated by the Jedi rather than by an act of the Senate or by an overt hostile act undertaken by the Separatists against the Republic or a member state which did not intend to secede. The government may not like it, but it is not treasonous to refuse to support it in acting illegally against its own member states - though it may very well be unwise, even if the government cannot afford to retaliate for the perceived disloyalty immediately.

As to whether or not the existence of neutral systems breaks continuity, it's hard to say; I certainly don't recall any mention of neutral systems in the Prequel Trilogy, but on the other hand the Military Creation Act (MCA) appears to have been contentious and only a very naive observer could believe that the army which invaded Geonosis could have been raised in the time between the passage of the MCA and the army's arrival over Geonosis; furthermore, the Jedi do not appear to have the express authorization of the Senate for their actions on Geonosis, nor is it clear that they even informed the Senate of what they were doing until after it was far too late for the Senate to do anything about it. Given that, I can see where 'neutral' Republic member systems might come from and where they might have legal basis for their neutrality.