how is there a system of checks and balances to keep the voters from appropriating everything
The US doesn't have one of those.
Literally. The people of the United States can, without consent of Congress or the President or the Supreme Court, do everything you just said, via the amendment-convention process (Congress does not have the option to deny a convention when at least 2/3 of states call for it). Once the convention's underway, even state legislatures can be bypassed by conventions in the states.
Thus it's possible to get amendments into the US Constitution without them ever having been voted on or even considered by Congress or the state legislatures. And if you're worried about a populist wave "appropriating everything" in other countries, why not worry about it in the US? It's just as possible.