Is there any way in Eagle to disconnect nets like the attached picture, to reduce clutter on more complicated designs?
Yes, make a short net coming off the pin. Name it something meaningful to that pin. Do the same on another pin that you want a net connected to. Name that pin/net the same as the first one you did. It will ask you if you want to connect "N$x" to "yourName." Hit ok. now you will see on the board view that there is a connection between the pins and you can route it.
Don't forget to add a label to the net so you what what it is!
codlink's method is what I use.
If you want to take it one step further and get the flag style labels, click info ("i" icon), then select the label in question. check the "Xref" box, and the label transforms from floating text to text in a box.
That's exactly what I wanted. Thanks to you both.