Just to be clear, I don't think CMs are more democratic in a systematic way. In fact, only a few monarchies around the world, even those based in constitution, are. The vast majorities are constitutionally anchored despots.
However, I do agree with Red that within the few western monarchies that actually work, monarchy is a very discussable and changeable thing. Not everybody can become King, true, but the majority of people currently feels that is fine. In fact, I would argue that the most influential thing making NL/DK/SV more democratic is in fact nit a monarchy, but the electoral systems in those countries which guarantee: one man, one vote, equal representation.
Trix was a great monarch because she knew her limits. Had she stepped out of line like her parents, I'm sure we'd be discussing a Dutch president now.