The generic remote cannot control channel strip items (other than eq, which was implemented before the channel strip existed). It should be able to.
You can see from the way EQ is set up in generic remote that you should be able to control the other channel strip items, almost as though the ability is there in the background but it is not visible in the generic remote editor (look at how EQ parameters are named).
The remote control editor supported controllers (e.g. Mackie Control) can control the channel strip.
A channel strip is much more useful if it is always easily available from a remote controller.