The auto-size feature was apparently broken a while back ... probably right around version 3.0. It was only somewhat recently brought to my attention that it was broken. It was unknown to me for a few years, and several versions. That does make me question its value.
I actually fixed it for the next version. But it's still a clunky kind of way to resize the screen. Instead, I'd recommend (a) using a layout that is intended for your display size (or closest to it) and (b) using layout scaling (preferences button on the Layout tab).
Quick question: We hook up the computer via HDMI to an external monitor. Is there a way to configure the external monitor to display the main tourney screen only and not the settings window when we’re busting players out or consolidating tables etc?
Preferences tab -> Display Management category
Everything you need is there.
In the next version, the main window can go full-screen and back to window mode by pressing F11 (like your browser). So you'll be able to just drag the main window to the display you want, hit F11, then hit ESC to open the Settings window. There's a preference to tell the TD to open the Settings and other windows on a specific display, so your game window can be on one and your Settings and everything else on the other, with no need to tell the application about your displays or use controls to move the game window around.