The Danes are smart, but history have repeatedly proven that people are deceptive, even the seemingly trustworthy ones that hands out promises for votes.
Europeans don't get to blame the US for this. Chat Control is being widely pushed. The Europeans get to take responsibility for their own authoritarianism.
The ruling elites on both sides of the Atlantic have been pushing for something like this for a long time now. It's not an either-or - they are mutually reinforcing.