He was the only male left in the imperial family when Caligula was assassinated...the Guard wanted to keep their jobs and privileges so they wanted the system created by Augustus to continue. The Senate debated restoring the Republic...the Guard did not want that.
VDH’s comparisons are at odds with your facts. It appears to me you are in command of the history whereas VDH is writing fluff more or less. His comparison makes for interesting thoughts but starts off shaky as the United States is still a Republic whereas Claudius did not have that going for him.
Regardless, there are more important fish to fry.