Not sure I completely follow, but let me throw this out there: there is a distinct difference between the displaying the "Tournament" screen versus displaying any of the built-in screens, like the "Player Rankings" screen or the "Seating Chart" screen.
When you display a built-in screen, by either pressing one of the big buttons under the title on the Settings window, or by pressing a hotkey (F3, F4, F5, F6 by default), the Game window will display that screen for an indefinite amount of time. In other words it will not change by itself. It will display the selected screen until you tell it to display something else.
When you display the Tournament screen (F2), you are telling the software to display in the Game window the screens that are determined by your Screen Sets. Note that this could include displaying built-in screens. The difference is that the screens will automatically change based on time and tournament conditions.
So as an example, let's say you've create a single custom screen called "My Screen". Assume you have only the Default Screen Set and it looks like this:
My Screen (30s)
Seating Chart (10s)
Player Rankings (10s)
If you press F2, in the Game window you'll see "My Screen". After 30 seconds, it will automatically change to display the Seating Chart. After 10 seconds it will automatically change to display the Player Rankings screen. After 10 seconds, it will go back to display "My Screen". And the rotation repeats and continues.
If you press F4, the Game window will display the Seating Chart. It will never change from this screen until you instruct it to change by selecting a different screen.
So, instead of thinking of pressing F2 to get to the "Tournament Screen", think of pressing F2 to "start screen rotation", or "run your screen sets".