When I was testing the Z5 for my book, I noticed this same problem you are referring to. That is, when the release mode button on the camera back is hit, the Single/Low/High/Self-time menu should be displayed in the rear monitor - unless your Monitor Mode selection is specifically "Viewfinder only". All the other settings (Automatic display switch, Monitor only, and particularly Prioritize viewfinder), the Release mode should show on the rear monitor, just like the menu display upon MENU button hit. That's how my Z7 is doing. I do not have the Z5 anymore, so I cannot test it again, but I believe the release mode did not show if the Monitor setting is "Prioritize viewfinder" that I normally use. In my opinion, this is definitely a software bug on the part of Nikon, and they should fix it right away... There is no reason that the Z5 alone should behave differently than Z6/Z7 and Z6ii Z7ii.I just telephoned Nikon Belgium and a very friendly lady tried to help me out. She was surprised and at first didn't have an answer. She did some research but found nothing that could shed some light.
After a few minutes she asked me: "When you look through the viewfinder and push this button...is something happening?" And yes...there it was...the button functioned. As I put the camera on 'viewfinder priority' it is only shown into the viewfinder, not on the monitor. The timer/release mode button is only shown on the monitor when the monitor is activated through the mode selection of the EVF/monitor. Seems logical you might think but all the other buttons function and are shown on the monitor no matter what mode you are in. But not the time/release mode button.
Mystery solved...I have peace of mind now. Thank you!