Why don't we raise the speed limit if that's how fast everyone wants to go?
Because then everyone would drive above that new limit instead.
I'm not so sure about that. If the speed limit in my neighborhood was 80mph I don't think I'd be going that fast much less above it. If the nearby interstate reduced the speed limit to 25mph I don't think I'd naturally be driving it at 35-40mph as you state.
In my experience, it's more a thing that I think many people disregard the speed limit for the most part and have a certain speed they feel is natural and comfortable for them to drive on a given road in a given situation.
Going through the neighborhood? Some people think 15mph, some think 25mph, some think 30mph. See some kids playing on the side of the road? Most people generally would slow down a bit. Roads are tight and narrow? Well now I have to be a bit more careful in this area so I need to slow down even further.
In my experience the psychology trick has everything to do with the road, what's around it, the condition it's in, etc, and less to do with the sign itself.