Ok, I've gotta take issue with this.
Dutch people litter. All the time. People rarely seem to clean up after their dogs. People place their trash bags out on the curb (where there aren't underground bins) far earlier than they should, resulting in trash-strewn streets and fat, obnoxious seagulls. It's a mess. Beer cans left on bridges and park benches. Energy drink cans tossed to the side. Cigarette butts strewn carelessly. Firework refuse absolutely fucking everywhere a few weeks either side of New Years. Bikes left as litter.
There's always someone who comes along with a street vacuum, or a team of people going along with bags and pickers. It seems to me that people have little respect for not littering because it's always someone else's problem, and there's always going to be someone cleaning up after you.
So if by "we" take care of public spaces you mean "lots of people get paid to clean up others' carelessness," sure, but in six years I've seen little to suggest that not leaving trash just anywhere is strong tenent of Dutch culture.
My hunch is that urban areas of the NL are essentially all developed and man-made, in that every street, sidewalk, tree, bush, patch of grass is planned and raw, untouched nature is relatively less accessible and visible. Therefore people perceive this urban "fabrication" with its attendant cleaning staff as less precious, less worth keeping clean than some primeval forest or national park.
I could be wrong, of course, and this is all just my own perception. But I really fail to recognize the cultural cleanliness that you say embodies the Dutch.