There is no such thing as a totally waterproof boot. Believe me, in my decades of wilderness backpacking I've tried it all. At best, the inevitable is only kept at bay for a bit longer. The US Military recognized this during the Vietnam war. My boots were fabric tops and leather bottoms with holes in the leather at the insole level. The holes drained the water that would soak into the boots when slogging through standing water in the jungles, or through rice paddies and shallow ponds.
Trail running shoes DO dry out pretty quickly. Boots will stay wet a lot longer than the shoes will. I have used trail runners on hikes throughout the Cascades, for weeks at a time, where rain was a given and soggy, puddled trails were inevitable. Yeah, the shoes (New Balance Leadville) would get wet, but paired with the smartwool socks I use, kept my feet comfortable. No blisters, and things dried out overnight.
I also carry an extra pair of lightweight insoles to swap out if the wet insoles aren't quite dry enough. And of course, extra socks. If I expect rain, I will apply a heavy coat of something like Body Glide, or Bonnies Balm to my feet to help keep them protected from pruning from excessively prolonged exposure to water. Of course, periodic breaks to allow for removing shoes and to dry off feet a bit is a good strategy.
For me, the conversion to trail runners about five years ago, was eye opening. Without the extra weight on my feet I am able to go farther with less fatigue.