I'd say it's not out of character for him. He's always been a bit of an unpopular asshole.

On the other side, it was for national TV, so if they let it slip that one time, that'd be newsworthy itself and impossible for them to gain back. So he had to do it for the sake of the whole institution.

While there are groups being against the monarchy, the general consensus is that part of their functions are irreplaceable by paid or elected officials. The Danes are pro-monarchy because the alternative is worse. They have a role to play in upholding morality and uniting the people which politicians never can or should do.

The new king is well liked by most people. They also demoted the titles of his brothers children recently which goes to show that they understand that "being born into it" isn't enough. Along with some out of the ordinary moves by the previous queen, they seem to understand that they serve the people and not vice versa.