You won't have plymouth in LFS unless you built and installed it. My guess is you didn't, because it isn't in BLFS, not even in the systemd version. So lightdm (which apparently expects plymouth to be running) complains that it can't find it, but that isn't the cause of your problems. It's a red herring.
As you can see, lightdm starts up the X server normally after kvetching about plymouth. Then something goes wrong. The server stops with an error. To discover what that error was, you need to look at /var/log/Xorg.log.0.
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