The signatories to the Declaration avoided any reasoned theological dispute, as among adherents to different nuances of Creation--or duties to God--by referring to the "Creator," as such. Whatever your theological view, the principle--i.e. that the legitimate function of Government is to secure the rights endowed by the Creator, to man in a natural state--is not dependent upon one's position on theological disputes as to the Creator's relation to man post Creation.
The compact theory of Government, as a rational consent to a particular Government, in common interest, is not dependent upon any of the theological questions that divide man in the pursuit of ultimate truths.
I would suggest that your query, here, is a "red herring" as to the elemental point under discussion.
Again, please read the document in context (Declaration Of Independence). It will provide many useful arguments against the endless succession of abuses that originate in slogan spewing Leftists, who count on Americans being ignorant of the underlying issues.
My contention in this thread is that the DoI is a founding document that does not have any force of law or expression of rights beyond what was in contention at the time with king George. They expressed their frustration with the abrogation of their rights as Englishmen under the rule of the crown.
As to the DoI providing arguments against leftest arguments and goals it does not directly. The DoI contains the arguments against the English crown, not against our current party of traitors. You can push that argument at them forever and they will laugh at you. They pervert the words in the DoI and the Constitution to mean whatever they want in their pursuit of tyranny.
We need to either have representatives who will address all of these issues based on constitutionality and the democrat party’s disregard of the Constitution or we, as defenders of the Constitution need to issue a new DoI that declares that the democrat party and our RINO reps no longer represent us and that they engage in criminal actions that are contrary to Constitutional law. That declaration would have to be backed up by actions that will remove them and restore the Constitution, the Republic, and the rule of Constitutional law.
Since the first alternative will never happen, the second seems to be the only viable path. Do I believe that enough support could be brought to bear on that path? I don’t know and a lot would depend on who is advocating that path. Who knows, the original DoI signers were a small group who struggled and sacrificed for years to achieve the goal with on and off again popular support.