Turns out, to enable the touchpad, one has to copy a reference Xorg config snippet:
cp /usr/share/X11/xorg.conf.d/50-synaptics.conf /etc/X11/xorg.conf.d/50-synaptics.conf
In addition, one can modify the file to set default options for tapping and scrolling.
* * *
So in fact the "problem" is that touchpads is nowadays handled by libinput, not by synaptics. This is why xinput still lists the device, but synclient cannot find it.
The touchpad properties can also be controlled using xinput, via `xinput list-props` and `xinput set-prop`—however, personally I found the libinput-touchpad-driver to be way less versatile than the synaptics driver, and not suitable for my purposes. Your Mileage May Vary.
If you simply want to change a specific single setting, you may want to look into `xinput`, before messing with your Xorg config.