False - they get unelected, overthrown, beheaded, or a host of other things when the people don't like those representing them.

You're really agreeing with everyone here. The "monopoly on violence" we're discussing comes in the context of a theoretical social contract[1], discussed by various philosophers. The social contract is a philosophical framework for deciding whether a government is legitimate or not. Under the contract, the government gets a monopoly on violence, and citizens get protection and support from the government. No one is suggesting that it's impossible to behead the king, just that this is not in the terms of the contract.

[1] - https://en.wikipedia.org/wiki/Social_contract

To refuse the social contract is to accept natural law (law without an organized government with a monopoly on violence). If too many people decide the social contract is not in their favor, they will do this by noncooperation or overthrowing the government. Under natural law, people are not prevented from building shelter, freely gathering food, etc. This is why people believe it is unjust and illegitimate to hold property rights above basic needs. Property (or at least more property than you can personally protect) is essentially a construction of the government, and freedom to gather and grow food is included in natural law. People who are deprived of important natural freedoms are likely to refuse the contract.