Unfortunately, direct democracy would reduce the power of current legislators, so it is unlikely that our political system will change.

The likelihood of Swiss-style direct democracy being instituted in the United States is low. From the Federalist Papers onward, protection from the "tyranny of the majority" of minority interests has been been fundamental to the U.S. political system. The system of checks and balances was primarily instituted as a prevention against a tyrannical legislature, not the Executive. There would be intense pushback from both major political parties, political science academics, think tanks, and lobbyist firms.

Using current events as an example, I don't think it would be out of the realm of possibility for a national referendum to pass that called for mandatory citizenship checks of people who appeared to be not born in the United States. Direct democracy works for homogenous populations, but I doubt it is the right fit for the United States.