All the classifiers are automatic. As you have seen, some are better than others. Sometimes there is a manual pass afterwards to clean up. Sometimes the classifiers are set too conservatively in order to "make sure" no non-ground (veg, building) points are left on the ground surface. The goal is to meet the error spec at the average specified post spacing, and most agencies are just interested in their 2 foot contour products, not knolls, cliffs and small reentrants.
I often re-classify datasets myself in order to gain back more of those ground points that have been incorrectly rejected. There's a compiled version of the classifier I use
here. There are others around. I think LAStools has one? Read the Evans paper there for more details. This one is designed specifically for forested areas. I usually set the scale to the avg post spacing and the curvature threshold to 0.35 or 0.4. I've played extensively with both terms and with the surface model,trying different ones other than the TPS. I use a simple Delaunay Triangulation, which was just as good and much faster. Ed Despard has modified the TPS in the MCC C code here to run much, much faster. I translated the MCC into IDL and use that. I also have a python version that James Scarborough translated.
When re-classifying with a less conservative cut you will sometimes have some veg accidentally left on the ground surface, so you have to be wary of "errors of comission."
Give it a try on this set and see if it helps.