Like anything else, it depends.

If there is only one crossing I'll try to keep my feet dry. But if it's a hot day and the trail ahead is dry I find the water refreshing on my feet. Those conditions existed a couple of weeks ago when I was in SEKI and I purposely walked through part of Hamilton Lake. My feet were dry again in an hour. I use thin nylon socks and that really helps.

I'll agree that dry socks quickly become wet socks when worn under wet shoes.

If I'm facing multiple stream crossings and/or it's raining I don't even try to keep my feet dry. I think resistance is futile. If I feel a hot spot coming on I tape it with Elastikon. Great stuff. It stretches with your skin and the adhesive is waterproof. I usually leave it on for several days. Injinji socks really help too. Last year on a long hike my feet were wet all day every day for nearly 4 days and I had nary a problem.