Actually I already tried to decipher this LAYER thing from the source code, unsuccessfully. I think it just doesn’t work in the worksheet.
There exists a simple workaround, though. Just create a text item on the board and copy it in all layers.
Although I have to admit it’s not much easier than writing the layer names yourself.
You could report this as a bug. The variable is documented in the worksheet (drawing sheet) editor and the variable exists in the source code for drawing sheet, so I wonder if it should work as you expected. Maybe @JeffYoung who has worked on text replacement can comment on this? For reference, here is the other thread about that variable: How to use keyword 'LAYER' in the worksheet editor