Yes, they should go down. They are not "powered" down, i.e. they don't go slowly. They go down by spring force. It's like when you raise them up they go against a bit of spring force and then latch, and when you push the button they unlatch and spring down. On my W124's there is a momentary pause before they go down, maybe a second or so.

I don't think the shade should affect things. Push the headrest release and see if you can then wiggle the headrests and get the latch(es) to unlatch. They might just be stuck if they've not been operated for a long time.

Rgds,
Chris W.