What GP means is that US is no longer a republic. Unifying aspect of the Obama, Sanders, and Trump campaigns is tapping the desire of the American public to make it one again.
No, what GP means is that the US is no longer a democratic republic (which is, as the term "democracy" is generally used today, a form of "democracy", rather than an alternative to it.) The complaint GP makes does not contrast with a republic in any way, as all a Republic means is that it is a system of government where government isn't by (or exercising powers derived, at least in principle, from) a monarch.
They define a republic to be a government of laws, and not of men. If this definition be just, the British constitution is nothing more nor less than a republic, in which the king is first magistrate. This office being hereditary, and being possessed of such ample and splendid prerogatives, is no objection to the government’s being a republic, as long as it is bound by fixed laws, which the people have a voice in making, and a right to defend. An empire is a despotism, and an emperor a despot, bound by no law or limitation but his own will; it is a stretch of tyranny beyond absolute monarchy. -- Novanglus, February 6, 1775
The opposite of republic is not monarchy, it is tyranny. The only way out of tyranny for the U.S. is, by definition, to restore the rule of law, which the GP seems to clamour for.