Pro/ENGINEER drawings rely on the parts and assemblies to generate views. So if you are going to copy the process drawing, you have to copy the process assembly. To maintain the associativity, you also have to copy the design assembly.
You are right. It makes sense to be able to copy the entire design, including process assemblies and drawings. However, Pro/INTRALINK doesn't automatically recognize the link between the process assembly and the original assembly when Duplicating Objects. This may be a bug or intended functionality. You would have to log a call with tech support to be sure.
There is a work-around. Create a manual link between the process assembly and the design assembly by selecting Object > Modify Relationships... Attach the design assembly to the process assembly and select OK.
Then you can highlight the process assembly and select Duplicate Objects. Now it will allow you to copy the design assembly, process assembly, and all associated drawings.
If you only want to copy the process drawing (not all the other drawings), highlight it and select Duplicate Objects. Then you can turn off the Include Drawings option.