It runs roughly every hour (with a few additional express trains) on the weekend, very similar to the rural trains you describe in Denmark.
My point is that you can't just show up at a commuter rail station and complain that you have to wait a while at a bare-bones platform. It's built for people who take it at the same time every day, commonly driving to the station.
The "Lokaltoger" on Zealand run every half hour on Saturdays, and every hour on Sundays, with the stations at 40-70km from Copenhagen. The settlements are mostly around 1-2 thousand people, although here we can include children and the elderly as potential users as it's not expected to drive to the stations.
If I use public transport every day during the week, then when going to similar places at the weekend I will certainly complain if the trains are infrequent.
http://www.caltrain.com/schedules/weekend-timetable.html
http://www.lokaltog.dk/media/1179/920r-frederiksvaerkbanen.p...
Outside the USA, public transport may run a little less often on a Saturday or Sunday, but it's still useful for people to use for shopping, going out and so on.
For example, in Copenhagen the trains run every 10 minutes rather than five, and the rural trains are every 60 minutes on a Sunday, rather than 20. I think the metro is the same at the weekend as during the week.