To clear assets you need do this instead.
Select a blank area on your route. About 2 or three squares is good for removing trees...
In the options, you need to uncheck things so you don't use those options - some weird results may occur as you've found out.
In your case, uncheck the track (splines) icon and terrain. This will prevent the paste option from pasting no track and affecting the terrain. Since you are pasting no objects (trees), you'll be deleting them instead.
In the end, you may need to get out the chainsaw in a few places.
I used a bit of bother here for this area.