Being a monarchist doesn't necessarily mean being a royalist (or a royal family fan), especially in bigger countries like the UK. In the specific case about the way King's Day is celebrated in the Netherlands (more like a Carnival than a royal celebration properly), I think it is simply not very compatible with upper class British culture, which is more circumspect.
I also don't think there is a clear correlation between educational attainment and support for the monarchy. There are highly educated Britons (Nobel prize winners, university Fellows, etc.) who are monarchists and, vice-versa, there is also a smaller number of highly educated people in the UK who are republicans.
Perhaps it has to do more with political leaning than education, i.e. Conservative voters being generally more pro-monarchy and left-wing Labour voters or nationalists being more pro-republic. Generally speaking, however, a sizeable percentage of the population is just neutral or doesn't have an opinion on the question of monarchy vs republic (less so in the UK than in other Commonwealth realm like Canada).
I don't know about Belgium or the Netherlands, but in Sweden, Spain and the UK , there are regular national polls that measure support for and opinions about the monarchy and the Royal Family. Those are scientific polls taken with random samples that give a pretty accurate point estimate of approval rate for example within the poll's margin of error and confidence intervals.
We don't need millions of people to cast a vote in the secrecy of the ballot box to measure support for the monarchy or the republic, or for that matter to measure the approval rate of a president or say which party is most likely to win the next election and by what margin. That is what polling is designed to do.