It may be different for you, but I have a similar issue.
I think it is because I have FileVault running on my Mac. FileVault doesn't allow a full boot up before first login so the DisplayLink software doesn't launch until you have completed the first log in.
The solutions I have come up with so far is to either ssh into to my macmini, use VNC, or use USB-C, or HDMI to enable the first log in then it is fine.
My alternate solution is to disable FileVault. You can check if FileVault is active using terminal
/usr/bin/fdesetup status
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