Unamendability doesn’t only protect democracy, it also protects outdated principles. The American Constitution protected slavery. In other parts of the globe, unamendable constitutions have protected the oppression of minorities.
These unamendable constitutional elements foster division and lead to division in society and human rights abuses. Under a fully democratic political system, the electorate needs to have the ability to remove these abhorrent elements from their constitution. If that means their democratic principles are left slightly more exposed, then so be it; for it is only by making every element of the constitution amendable that democracy can be fully guaranteed.
The claim that unamendability protects democracy better than an amendable constitution is also highly dubious. If the majority of the electorate demands constitutional change but the political and legal system offers no avenue for it, the population is far more likely to take up arms and pursue constitutional change through violent means.
Therefore, rather than protecting democracy, constitutional unamendability hastens its downfall in highly engaged, politicized societies.