This claim is ahistoric. The ancient Athenians, the inventors of democracy, would reject it because they used popular elections in only a limited number of cases. Their suspicion was that popular elections were tools of oligarchy. Instead they preferred sortition, selection at random, to give rule of the people. They punished abuse of public power severely.
In the context of American thought, Federalist No. 10 goes into exacting detail as to why the proposed government was a republic and not a democracy. If a republic were merely form of democracy, then the entire document would have been a waste of time. Instead, this was a point of serious debate. https://avalon.law.yale.edu/18th_century/fed10.asp
Hence it is that such democracies have ever been spectacles of turbulence and contention; have ever been found incompatible with personal security or the rights of property; and have in general been as short in their lives as they have been violent in their deaths. Theoretic politicians, who have patronized this species of government, have erroneously supposed that by reducing mankind to a perfect equality in their political rights, they would, at the same time, be perfectly equalized and assimilated in their possessions, their opinions, and their passions.
The U.S. constitution is explicitly anti-democratic on several points. Judges are appointed rather than elected and serve for life, intentionally intended, although admittedly with limited success, to remove them from partisan pressures and the fickle passions of the day. States have unequal representation in the House. Large states and small states have equal representation in the Senate. The president is not elected by popular vote but by a select group of electors. Executive, legislative, and judicial are co-equal; one may not compel the other even with an appeal to some election. Even a unanimity of voters may not pass certain legislation.
The very purpose of a Bill of Rights was to withdraw certain subjects from the vicissitudes of political controversy, to place them beyond the reach of majorities and officials and to establish them as legal principles to be applied by the courts. One’s right to life, liberty, and property, to free speech, a free press, freedom of worship and assembly, and other fundamental rights may not be submitted to vote; they depend on the outcome of no elections.
Brown v. Board, https://tile.loc.gov/storage-services/service/ll/usrep/usrep...
This compulsion to torture both language and history to apply the blessed label democracy to forms of government that do not meet the definition is puzzling. Call things what they are. Democracy is not a worthy end in itself. Majoritarianism and utilitarianism can be highly problematic and downright evil.