David, the short answer is yes, the score editor can be forced to do what you want. That said, you'll need to spend some time with either the manual or the excellent Logic Score Guide written by Johannes Prischl - several versions old, but still completely valid, as the score functions have not changed in a long time.
http://prischl.net/LNG/LNGFramesets/LNG_Fr3.htm
I have been able to learn how to make Logic do what I want it to virtually all the time, just through trial and error (and the occasional question on this Forum) and can now use it for all my scoring requirements. (Caveat: I work in commercials, so my pieces are rarely longer than a minute)
In your posted example, you need to select a different Score Style for the Bassoon part, and use the Voice Separation Tool (looks like two arrows, one up on down) to divide the left and right hands of the piano part. There are Interpretation, Syncopation and Quantize settings that need adjustment too, as grooveiron described. HTH
Sonny Keyes
Ricochet Audio
Toronto