The problem with this argument is that it assumes that the issues people vote on don't matter if the politician they vote for does something undemocratic. If a politician decides to do something such as having an extramarital affair, that is seen as undemocratic. But why should that, which is a part of the politician's personal life, affect how and why people vote? If he stands firmly on the right side of an important issue that people feel strongly about, why should it matter how he acts outside of office?
Democracy isn't undermined by allowing the people to vote based on "their own self-interests," because it is in everyone's best interest to come to decisions about important issues, such as the aforementioned abortion, gay marriage, healthcare, and economy. Today voters are voting based on dire issues such as Covid; is it not in everyone's best interest to pick politicians who will solve the Covid crisis quickly and efficiently? Does it matter if a politician does something "undemocratic" if they also make a decision which ultimately saves millions of lives?
Democracy works because what the people want is what matters most. If we expect everyone to vote based on how undemocratic a candidate acts, instead of focusing on the real issues at hand, then nothing will ever get done.