The Republic form of government isn’t what allows that – what allows it in this case is the First Amendment (free speech, and right to petition for redress of grievances). Lobbyists come from businesses, special interests, citizens’ groups, etc., petitioning on behalf of citizens and interests. Some of it is terribly over-the-top, but it’s collateral damage from our much-needed right. Throw out the First Amendment rights to avoid lobbying, and you throw out the baby with the bathwater.
On the other hand, since I know that the Asker leans right, what I will submit for consideration here is the recent SCOTUS decision removing limits of corporate funding of candidates’ campaigns. By definition, this is allowing large corporations (a special ”$ cash rich $” segment of our society) to buy the government that they want. Since large corporations are “people” only in legal terms, this is an even more egregious abuse of the First Amendment’s guarantee of Free Speech – yet it was a right-leaning court’s decision, and is likely to lead toward more candidates from the Right gettting elected, since Big Business generally supports the Right (and the Right supports Big Business in return). It’s possible to muzzle Business, though, without also taking down ordinary citizens’ right to petition the government.
(Even in a direct democracy – which is impossible on a Federal level in a society this big and complex – those with deep pockets could buy votes. They would just have to spread it out to more people, in an Astroturfing kind of way.)