If inability to remove the Consul (or Dictator) via from office via elections is what qualifies as the death of the Republic, then it died with Marius, Sulla, or Pompey. Laying the death at Caesar's feet is ridiculous.
Caesar was popular. He was murdered by a coalition of unpopular and disaffected Pompeiians and noveau riche both enabled by Caesar's clemency and largess. He was accorded his honors, powers, and cult status legally. If your notion that a Republic isn't a Republic because popular representatives can't be voted out (because the people don't want to vote them out), then Republics and Democracies don't exist at all and all you have is rhetoric.
Caesar was popular. He was murdered by a coalition of unpopular and disaffected Pompeiians and noveau riche both enabled by Caesar's clemency and largess. He was accorded his honors, powers, and cult status legally. If your notion that a Republic isn't a Republic because popular representatives can't be voted out (because the people don't want to vote them out), then Republics and Democracies don't exist at all and all you have is rhetoric.