How can I enable USB debugging on my HTC One with Android 4.4?
As it turns out, the issue was that my phone was no longer being recognised as a composite USB device by Windows, meaning only one of ADB, MTP or HTC Sync was capable of running at a given time. After the update, MTP was always recognised first. This explains both the 4.3 issue of MTP and ADB conflicting, and the 4.4 issue of no ADB.
The solution was provided by HTC's support website:

Step 7. If you could see your phone as the portable device on your PC, but there is still no &quot;Android USB Device&quot; in &quot;Device manager&quot;, please try to manually update the driver for your phone.

Expand &quot;Portable Devices&quot; in &quot;Device Manager&quot;
Right click on your phone's name or &quot;MTP USB Device&quot;
Update driver software
Select “Browse my computer for driver software”
Tap “Let me pick from a list of device drivers on my
computer”
Don’t check “Show the compatible hardware”
Select Standard USB main controller
Select USB composite device in the right hand side list as the screen shot below and then click Next
After you click Next, if there is Warning message window pop up, please choose Yes.