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:
<service block="true" log="true">
<action name="android.service.chooser.ChooserTargetService" />
Now you have to put it on the phone and copy it into the Intent Firewall directory. On most devices that would be
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.