Removing a contact from "Direct Share" panel
It turns out that - for rooted people - there is a way to completely turn off Direct Share!

For the short version skip to the bottom of this answer.

In my search for a way to stop this stupid feature from bugging me I had a look at how the feature is actually implemented in apps by developers. It turns out that they register a service with an intent receiver (read: event listener) and when I realized that, I started looking for a way to block certain intents from broadcasting through the system. Well, Android actually has that built in. It's called the Intent Firewall. I quickly figured out how to create a rule for it to block apps from receiving the "Hey are there any apps that want to bug the user by putting a couple of Direct Share buttons under their thumb as they're about to tap something?"-event.

You just create a simple xml file like and drop this into it:

&lt;rules&gt;
  &lt;service block="true" log="true"&gt;
    &lt;intent-filter&gt;
      &lt;action name="android.service.chooser.ChooserTargetService" /&gt;
    &lt;/intent-filter&gt;
  &lt;/service&gt;
&lt;/rules&gt;


Now you have to put it on the phone and copy it into the Intent Firewall directory. On most devices that would be 

/data/system/ifw/


I simply used Root Explorer to copy the file there (no chmod required), tried to share something that caused Direct Share targets to pop up and BOOM! Nothing!

I recommend to do some background reading (see links above), especially about how the Intent Firewall works and how you can find the folder.

Happy not-sharing!