Don't look for the Federal BATF policy statement ceasing enforcement actions for portions of the NFA to affect anything in California.
California law does exempt a number of weapons from California statutes that would otherwise prohibit them if they are an "Any Other Weapon" under the federal NFA and are in the possession of a person permitted to possess them under the NFA (refer to Penal Code section 11710).
But the key provision here is whether the person is "permitted to possess them under the NFA." The recent federal BATF action only ceases enforcement actions. It does not amend, or in any way change the content of, the NFA. What the NFA permits remains the same as it was before the BATF action.
The NFA statute may become infirm when there is a final decision on the case that prompted the BATF policy change, or it may be amended by Congress, but neither has occurred as of yet.
Keep in mind that the general federal Statute of Limitations for felony offenses is five years. A person who acquires an "any other weapon" in violation of the text of NFA, and upon assurance of no prosecution by the current administration, could well find themselves charged under the next administration.