I don’t remember ever (since 90s) having references on the board.
Using Protel 3 for long time we used its silkscreen layer (have the advantage of following footprint when flipped top/bottom) for our documentation so we didn’t had it on PCB. When we needed to use silkscreen layer it was only for text for user like terminal block pin description (texts of kind: “GND” “+12V”, “IN1”, “IN2” and so on). In such case I placed those texts on one graphic layer and sending PCB to manufacture informed them to use this layer as silkscreen.
When moved to KiCad in 2017 I found the other layer pair swapping layer with footprints (courtyard) and used it for documentation so now our PCB have silkscreen but containing only rectangles around footprints. I have references and values at courtyard layers to have them in our documentation.
How I place references you can see here: