I found it! I can reproduce the issue, it's not random, we can open a bug report. It happens if you focus another window with the mouse, than even though you can focus the window with touch, the button won't work there.

The button works this way:

1. Click on the header of the Console window with mouse
2. Touching the menu button works on the Console window.
3. Touching on any other menu buttons on different windows doesn't work.

The button doesn't work this way:

1. Click to any other window or just on the empty desktop with mouse
2. You can focus the window with touch, but the button doesn't work.

So in other words, it works only, if you focused the window with mouse previously. I can't see anything related to this in the logs. Please try to reproduce.

I'm still on Gnome 44, I would wait until I get Gnome 45, to see if maybe they fixed this.