Even with an unamendable constitution, voters keep their primary constituent powers.
Unamendable constitutions stipulate that no government may amend the constitution, but they cannot stop a government from drawing up a new constitution. Therefore, the voters retain their right to choose the constitutional elements that they want to live under but instead of amending an existing constitution, they would need to get their government to draw up a new one.
Because voters are able to keep their constituent powers, unamendable constitutions are compatible with democratic principles and have a place in democratic societies.
This argument also suffers from a narrow definition of democracy. Democracy is not merely the process through which we appoint leaders and make collective decisions. Democracy is also the process of guaranteeing individual rights to all citizens.
Building an unamendable constitution furthers democracy in this sense. By preventing constitutional tinkering, an unamendable constitution can ensure individual rights are protected from malign political actors forever, guaranteeing the survival of a fully democratic society.