Ok, in a fit of madness I took the Quad apart and detached the leg of the diode which goes to ground. I bent the diode up a teeny bit to break the contact, the other two legs are still attached. I figured it would be easy to push it back down again if the experiment failed.
(Is that the sound of Electrical Engineers crying?)
Here’s a screenshot of the same 2MHz square wave as before but without the diode:
As you can see the digital input now works a zillion times better.
I guess the next step is to remove it completely and look for a replacement ASAP. I think I can probably manage to solder a new one in place, it’s not that tiny.