You're confusing them again, which is why sneaking in priors with the term "people" is bad. "granted to them by the people" means "We The People", society, etc, not the people in control of the legal fiction.

> being proxies of their constituents

These proxies have no rights. People can exercise their rights as people. Or, they can create legal fictions with no rights. That legal fiction inherits no rights from the owners, which is where the analogy fails. There's no intermingling/combination/whatever to call "rights", because there's nothing.

Every action that legal fiction takes is a permission granted to it by society that continues to tolerate its existence. That tolerance is finite. When that tolerance ends, so does the legal fiction's existence. Unlike with people, there are no moral quandaries with revoking the privilege of existence.

Rights being inherent to people is an interesting but separate topic. We've generally agreed as a society that people have rights.