I went through the exact same nightmare after updating to 15.4.1 – HDMI was just gone. What finally worked was removing everything DisplayLink (not just the app, but also files in /Library/Extensions/ and ~/Library/Preferences/). After that, HDMI came back, then I reinstalled the latest DL and only used "Extend" – no weird app assign.
Be sure to reset NVRAM and SMC after every reinstall – it honestly helps untangle the display mess sometimes.
