lxdm installed --no-recommends is now allowing $PATH to be read from /etc/environment.
lxde-core/common installed with the --no-recommends option is now allowing $PATH to be read from /etc/environment, both with and without lxdm installed.
I'm not sure what happened earlier to cause me to think that there was a problem with lxde on its own. I wasn't using fancy login scripts to login without a desktop manager - just "startx". Maybe a stray configuration file that I hadn't removed was the culprit, though that's only a guess.
@Julien
These are the results of my own tests:
lxdm installed --no-recommends is now allowing $PATH to be read from /etc/environment.
lxde-core/common installed with the --no-recommends option is now allowing $PATH to be read from /etc/environment, both with and without lxdm installed.
I'm not sure what happened earlier to cause me to think that there was a problem with lxde on its own. I wasn't using fancy login scripts to login without a desktop manager - just "startx". Maybe a stray configuration file that I hadn't removed was the culprit, though that's only a guess.
Anyway, all is working properly now.