You can use this global keyboard shortcut to turn on the "Use keyboard navigation to move focus between controls" setting: Control + Fn + F7

Here how to test if it's on or off:

1. Open a new TextEdit doc.

2. Type some random gibberish.

3. Attempt to save the doc.

4. Hit "Tab" and see if anything happens. If the "Use keyboard navigation to move focus between controls" setting is off, nothing will happen.

Now, with the "Save" dialog still open....

5. Do the key combo: Control + Fn + F7

6. Hit "Tab" again — this time you should notice you're moving through the different UI elements.

Tab lets you move forward/down through the UI. Tab + Shift lets you move in reverse.

If you have System Preferences > Keyboard > Shortcuts open when you do the key combo, you'll notice the checkbox for "Use keyboard navigation to move focus between controls" toggle from checked to unchecked in realtime :)