It is always good to get an outsider's view of how it seems the U.S. is working. Ancient Greece may have been the last true Democracy, there is no country to lays claim to that today. The right of the people to vote on items such as peace, war, environmental or historical district, is the sole preserve of the elected representatives, hence a Republic. The nearest country that exists today, being closest to a democracy is Switzerland. There almost anything that can gather enough signatures becomes an item the people, not their representatives get to vote on. Britain has seen how leaving things to the people go, Scotland independence and Brexit, am not aware of any other countries in Europe that use the referendum in this manner.
What you call lobbying is protected by the First Amendment. It specifically allows for the people, to assemble peaceably, and to seek redress for their grievances against the government. That is lobbying. Big business is powerful, so are labor unions, so are environmentalists, gun owners, abortion activists on both sides of the equation, and the lists go on. In the U.S. sometimes business wins, sometimes labor, sometimes the environment, sometimes oil and gas interests, it is never for sure which side will come out on top, with a win. Non-government organizations are also extremely powerful and wage their own campaigns. No side seems to remain in the driver's seat for all that long. If big business were all that powerful the Environmental Protection Agency and the Clean Waters Act would never have come to pass.
Gerrymandering is a tough one. It was the first proposed amendment for what became the Bill of Rights. The states never accepted it, and recently the Supreme Court, perhaps using that, decided it was beyond their scope of authority to impose limits on it from a Federal basis, so it left it to the states,which some have tacked it and some have not.
The example of Kentucky was in a mail-in primary in which 70% of the voting was done by mail. There is only one county in Kentucky that has over 700,000 so it is a limited case. In any event the voting was the largest primary voting ever. So every vote still counted.
America also recognizes the rights of states, which is something that is alien to Europe. Each state has the same right to 2 Senators. The Senators represent their state not the people.
Hope this provides a different view of what is actually happening here.