With the tree at the side of the channel as a marker, I quickly realized that yes, the stream was steadily filling in the channel - the sinkhole must have just started overflowing.
I stood at the end of the little bridge for 10 minutes, and took a sequence of pictures. You can see the corner of the bridge in the lower left.
The sinkholes I refer to here are not the large sinkholes that sometimes collapse in Florida (and elsewhere). Although the process of formation is similar, in 'karst' topography, the rock here is very hard dolostone, and our sinkholes are really just cracks in the bedrock large enough to let water flow through - and sometimes form small caves. However, this is enough that streams disappear in these sinkholes and re-emerge halfway down the escarpment slopes, a widespread feature of drainage patterns around here. Perhaps I'll try to get a diagram to explain this sometime.